ΨΛ ΤΥ OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA fY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA 9se ? uNliEHSITY OF CUIFORHU LIBRARY OF THE UNIVERSITY OF CUIfORHlA UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA APOLLONIUS OF PERGA TREATISE ON CONIC SECTIONS. UonDon: C. J. CLAY AND SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE RIARIA LANE. ©Inaooh): 263, ARGYLE STREET. lLftp>is: P. A. BROCKHAUS. 0fto goTit: MACMILLAN AND CO. ^CNIVERSITT• Jniuppus 9lnlMus%^craticus,naufmato cum ejcctus adjViaMai/u UtJ ammaScrudh Geainetncn fJimmta defer, pta, cxcUnwimJIe aa coMXiXcs ita Jiatur^hcnc fperemus,Hominnm cnim veiiigia video. APOLLONIUS OF PEEGA TEEATISE ON CONIC SECTIONS EDITED IN MODERN NOTATION WITH INTRODUCTIONS INCLUDING AN ESSAY ON THE EARLIER HISTORY OF THE SUBJECT RV T. L. HEATH, M.A. SOMETIME FELLOW OF TRINITY COLLEGE, CAMBRIDGE. Ι^η\ονντ€$ τούί Ώ.νθα•γορ(ίοΐ'ί, oh πρόχαρον ηι> καΐ τοΓτο σύμβοΚον σχαμι ίαΐ βάμα, αλλ' ον σχάμα καΐ τριώβοΧον. Proclub. r-^^''^ . \ "^ Λ Uf 3 CAMBRIDGE : AT THE UNIVERSITY 1896 PRESS. [All Rights reserved.] /\b1 txfartl PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. MANIBUS EDMUNDI HALLEY D. D. D. ΟΓΤΜΡ '" UNIVERSITT^ PREFACE. TT is not too much to say that, to the great majority of -*- mathematicians at the present time, Apollonius is nothing more than a name and his Conies, for all practical purposes, a book unknown. Yet this book, written some twenty-one centuries ago, contains, in the words of Chasles, " the most interesting properties of the conies," to say nothing of such brilliant investigations as those in which, by purely geometrical means, the author arrives at what amounts to the complete determination of the evolute of any conic. The general neglect of the " great geometer," as he was called by his contemporaries on account of this very work, is all the more remarkable from the contrast which it affords to the fate of his predecessor Euclid ; for, whereas in this country at least the Elements of Euclid are still, both as regards their contents and their order, the accepted basis of elementary geometry, the influence of Apollonius upon modern text-books on conic sections is, so far as form and method are concerned, practically nil. Nor is it hard to find probable reasons for the prevailing absence of knowledge on the subject. In the first place, it could hardly be considered sui-prising if the average mathematician were apt to show a certain faintheartedness when confronted with seven Books in Greek or Latin which contain 387 PREFACE. propositions in all; and doubtless the apparently portentous bulk of the treatise has deterred many from attempting to make its acquaintance. Again, the form of the propositions is an additional difficulty, because the reader finds in them none of the ordinary aids towards the comprehension of somewhat complicated geometrical work, such as the conventional appro- priation, in modern text-books, of definite letters to denote particular points on the various conic sections. On the contrary, the enunciations of propositions which, by the aid of a notation once agreed upon, can now be stated in a few lines, were by Apol- lonius invariably given in Λvords like the enunciations of Euclid. These latter are often sufficiently unwieldy: but the incon- venience is gi-eatly intensified in Apollonius, where the greater complexity of the conceptions entering into the investigation of comes, as compared with the more elementary notions relating to the line and circle, necessitates in many instances an enun- ciation extending over a space equal to (say) half a page of this book. Hence it is often a matter of considerable labour even to grasp the enunciation of a proposition. Further, the propo- sitions are, with the exception that separate paragraphs mark the formal divisions, printed continuously; there are no breaks for the purpose of enabling the eye to take in readily the successive steps in the demonstration and so facilitating the comprehension of the argument as a whole. There is no uni- formity of notation, but in almost every fresh proposition a different letter is employed to denote the same point: what wonder then if there are the most serious obstacles in the way of even remembering the results of certain propositions? Nevertheless these propositions, though unfamiliar to mathe- maticians of the present day, are of the very essence of Apollonius' system, are being constantly used, and must there- fore necessarily be borne in mind. The foregoing remarks refer to the editions where Apollonius can be read in the Greek or in a Latin translation, i.e. to those of Halley and Heiberg; but the only attempt which has been PREFACE. ix made to give a complete view of the substance of ApoUouius in a form more accessible to the modern reader is open to much the same objections. This reproduction of the Conies in German by H. Balsam (Berlin, 1861) is a work deserving great praise both for its accuracy and the usefulness of the occasional explanatory notes, but perhaps most of all for an admirable set of figures to the number of 400 at the end of the book ; the enunciations of the propositions are, ho\vever, still in Wi^rds, there are few breaks in the continuity of the printing, and the notation is not sufficiently modernised to make the book of any more real service to the ordinary reader than the original editions. An edition is therefore still wanted which shall, while in some places adhering even more closely than Balsam to the original text, at the same time be so entirely remodelled by the aid of accepted modern notation as to be thoroughly readable by any competent mathematician ; and this want it is the object of the present work to supply. In setting myself this task, I made up my mind that any satisfactory reproduction of the Conies must fulfil certain essential conditions: (1) it should be Apollonius and nothing but Apollonius, and nothing should be altered either in the substance or in the order of his thought, (2) it should be complete, leaving out nothing of any significance or importance, (3) it should exhibit under different headings the successive divisions of the subject, so that the definite scheme followed by the author may be seen as a whole. Accordingly I considered it to be the first essential that I should make myself thoroughly familiar with the whole work at first hand. With this object I first wrote out a perfectly literal translation of the whole of the extant seven Books. This was a laborious task, but it was not in other respects difiicult, owing to the excellence of the standard editions. Of these editions, Halley's is a monumental work, beyond praise alike in respect of its design and execution; and for Books V — vii it is still tht• only complete edition. For Books i — iv I used for the most part the new Greek text of Heiberg, a schohir who has earned the undying gratitude of all who are interested in the history of Greek mathematics by successively bringing out a critical text (with Litin translatiun) of Archimedes, of Euclid's Elements, and of all the writings of Apollonius still extant in Greek. The only drawback to Heiberg's Apollonius is the figures, which are poor and not seldom even misleading, so that I found it a great advantage to have Halley's edition, with its admirably executed diagrams, before me even while engaged on Books I — IV. The real diHiculty began with the constructive work of re-writing the book, involving Jis it did the substitution of a new and unifonn notation, the condensation of some pro- j)ositions, the combination of two or more into one, some slight iv-arrangements of order for the purpose of bringing together kindred propositions in cases where their separation Λvas rather a matter of accident than indicative of design, and so on. The result has been (without leaving out anything essential or important) to diminish the bulk of the work by considerably more than one-half and to reduce to a corresponding extent the number of separate propositions. When the re-editing of the Conies was finished, it seemed necessary for completeness to prefix an Introduction for the purposes (1) of showing the relation of Apollonius to his pre- decessoi's in the same field both as regards matter and method, (2) of exj>laining more fully than was possible in the few notes inserted in square brackets in the body of the book the mathe- matical significance of certain portions of the Conies and the probable connexion between this and other smaller treatises of Apollonius about which we have information, (8) of describing and illustrating fully the form and language of the propositions ;is they stiind in the original Greek text. The first of these purposes required that I should give a sketch of the history of conic sections up to the time of Apollonius ; and I have ac- cordingly coiisidrn-d it worth while to make this part of the PREFACE. xi Introduction as far as possible complete. Thus e.g. in the case of Archimedes I have collected practically all the propositions in conies to be found in his numerous works with the substance of the proofs where given ; and I hope that the historical sketch as a whole will be found not only more exhaustive, for the period covered, than any that has yet appeared in English, but also not less interesting than the rest of the book. For the purposes of the earlier history of conies, and the chapters on the mathematical significance of certain portions of the Conies and of the other smaller treatises of Apollonius, I have been constantly indebted to an admirable work by H. G. Zeuthen, Die Lehre von den Kegelschnitten im AlteHnm (German edition, Copenhagen, 188G), which to a large extent covers the same ground, though a great portion of his work, consisting of a mathematical analysis rather than a reproduction of Apollonius, is of course here replaced by the re-edited treatise itself I have also made constant use of Heiberg's Litterargeschichtliche Studien ilber Euklid (Leipzig, 1882), the original Greek of Euclid's Elements, the works of Archimedes, the συναηωψ] of Pappus and the important Commentary on Eucl. Book I. by Proclus (ed. Friedlein, Leipzig, 1873). The frontispiece to this volume is a reproduction of a quaint picture and attached legend which appeared at the beginning of Halley's edition. The story is also told elsewhere than in Vitruvius, but Avith less point (cf Claudii Galeni Pergameni ΤΙροτρβ7Γτικο<; iirl τύχνας c. V. § 8, p. 108, 3-8 ed. I. Marquardt, Leipzig, 1884). The quotation on the title page is from a vigorous and inspiring passage in Proclus' Commentary on Eucl. Book i. (p. 84, ed. Friedlein) in which he is describing the scientific purpose of his work and contrasting it Λvith the useless investigations of paltry lemmas, distinctions of cases, and the like, which formed the stock-in-trade of the ordinary Greek commentator. One merit claimed by Proclus for his work I think I may foirly claim for my own, that it at least contains 'όσα 7ΓpayμaτetωBeστipap e^ei θ^ωρίαν; and I Ml PREFACE. should indeed be proud if, in the judgnieuL of competent critics, it should be found possible to apply to it the succeeding phrase, συντ€\(ΐ ττρος την ο\ην φιΧοσοφίαν. L•\st\y, Ι wish to express my thanks to my brother, l)r H. S. Heath, Principal of Mason College, Birmingham, for his kindness in reading over most of the proof sheets and for the constant interest which he has taken in the progress of the work. T. L. HEATH. MarcJi, 1896. LIST OF PRINCIPAL AUTHORITIES. Edmund Halley, Apollonii Pergaei Conicorum libri octo et Sereni Antis- seiisis de sectione cylindri et coni lihn duo. (Oxford, 1710.) Edmund Hallet, Apollonii Pergaei de Sectione Rationis libri duo, ex Arahico versi. (Oxford, 1706.) J. L. Heiberg, Apollonii Pergaei quae Graece exstant cum commentariis antiquis. (Leipzig, 1891-3.) H. Balsam, Des Apollonius von Perga sieben BUcher iiber Kegelschnitte iiebst deni durch Halley wieder hergestellten ctchten £ucke deutsch bearbeitet. (Berlin, 1861.) .T. L. Heiberg, Litterargeschichtlicke Studien iiber Enklid. (Leipzig, 1882.) J. L. Heiberg, Euclidis elementa. (Leipzig, 1883-8.) G. Friedlein, Prodi Diadochi in primum Eticlidis eJementorum librum commentarii. (Leipzig, 1873.) J. L. Heiberg, Quaestiones Archimedeae. (Copenhagen, 1879.) J. L. Heiberg, Archimedis opera omnia cum commentariis Eutocii. (Leipzig, 1880-1.) F. HuLTSCH, Pappi Alexandrini collectionis quae svpersunt. (Berlin, 1876-8.) C. A. Bretschneider, Die Geometric und die Geometer vor Euklides. (Leipzig, 1870.) M. Cantor, Vorlesungen iiber Geschichte der Mathematik. (Leipzig, 1880.) Η. G. Zeuthen, Die Lehre von den Kegel sehnitten im Altertum. Deutsche Ausgabe. (Copenhagen, 1886.) C/^lifOrnia^-- CONTENTS, INTRODUCTION. PART T. THE EARLIER HISTORY' OF CONIC SECTIONS AMONG THE GREEKS. PAGE Chapter I. The discovery of Conic Sections : Me- NAECHMUS xvii Chapter U. Aristaeus and Eucmp xxxi Chapter III. Archimedes jij PART II. INTRODUCTION TO THE COXICS OF APOLLONIUS. Chapter I. The author and his own account of the Ionics Ixyiii Chapter II. General characteristics Ixsxvii § 1. Adherence to Euclidean form, conceptions and language Xxxxvii § 2. Planimetric character of the treatise . xc\ni § 3. Definite order and aim xcviii Chapter III. The .methods of Apollonius .... ci § 1 . Geometrical algebra ci (1) The theory of proportions ... ci (2) The a])plication of areas .... cii (3) (iraphic representation of areas by means of auxiliary lines cxi (4) Special use of auxiliary jioints in Book vii. cxiii § 2. The use of coordinates oxv § 3. Transformation of coordinates .... cxviii L^ § 4. Method of finding two mean proportionals cxxv § 5. Method of constructing normals passing through a given point c.xxvii Chapter IV. The construction ok a conic by means ok tangents isxx xvi rONTENTS. PAGE I ΊΐΛΓΤΚη V. ΤΠΚ THREE-LINE AND FOUR-LINE LOCUS . CXXXviu I iiAiTKR VI. The constriction of a conic through five I'OiNTs cli Appendix. Notes on the terminology of Greek geo- metry clvii THE CONICS OF APOLLONIUS. THE CONE 1 THE DIAMETEK AND ITS CONJUGATE 15 TANGENTS 22 PROPOSITIONS LEADING TO THE REFERENCE OF A CONIC TO ANY NEW DIAMETER AND THE TANGENT AT ITS EXTREMITY 31 CONSTRUCTION OF CONICS FROM CERTAIN DATA . 42 ASYMPTOTES 53 TANGENTS, CONJUGATE DIAMETERS AND AXES. . 64 EXTENSIONS OF PROPOSITIONS 17—19 84 RECTANGLES UNDER SEGMENTS OF INTERSECTING CHORDS 95 HARMONIC PROPERTIES OF POLES AND POLAES . 102 INTERCEPTS MADE ON TWO TANGENTS BY A THIRD 109 FOCAL PROPERTIES OF CENTRAL CONICS . 113 THE LOCUS WITH RESPECT TO THREE LINES ETC. 119 INTERSECTING CONICS 126 NORMALS AS MAXIMA AND MINIMA I39 PROPOSITIONS LEADING IMMEDIATELY TO THE DETER- MINATION OF THE EVOLVTE 168 CONSTRUCTION OF NORMALS ISQ OTHER PROPOSITIONS RESPECTING MAXIMA AND MINIMA 187 EQUAT< AND SIMILAR CONICS I97 PROBLEMS 209 VALUES OF CERTAIN FUNCTIONS OF THE LENGTHS OF CONJUGATE DIAMETERS 221 INTEODUCTION. PART I. THE EARLIER HISTORY OF CONIC SECTIONS AMONG THE GREEKS. CHAPTER I. THE DISCOVERY OF CONIC SECTIONS: MENAECHMUS. There is perhaps no question that occupies, comparatively, a larger space in the history of Greek geometry than the problem of the Doubling of the Cube. The tradition concerning its origin is given in a letter from Eratosthenes of Gyrene to King Ptolemy Euergetes quoted by Eutocius in his commentary on the second Book of Archimedes' treatise On the Sp^re and Cylinder* ; and the following is a translation of the letter as far as the point where we find mention of Menaechmus, with whom the present subject begins. " Eratosthenes to King Ptolemy greeting. "There is a story that one of the old tragedians represented Minos as wishing to erect a tomb for Glaucus and as saying, when he heard that it was a hundred feet every way, Too small thy plan to bound a royal tomb. Let it be double ; yet of its fair form Fail not, but haste to double every sidef. * In quotations from Archimedes or the commentaries of Eutocius on his works the references are throughout to Heiberg's edition (Archimedis oprra omnia cum commeiitariis Eutocii. 3 vols. Leipzig, 1880-1). The reference here is ni. p. 102. t μικρόν 7* ίλίξαί βασιλικού σηκόν τάφου' δΐ7Γλασιο$ ίστω ' τοΟ καλοΟ δέ μη σφαΧίΙί δίττλαί' ίκαστον κώΧον iv τάχίΐ τάφου. Valckenaer (Diatribe de fragm. Eurip.) suggests that the verses are from the H. C. ^ XVUl THE ΕΛΗΙ,ΙΕΙΙ HISTORY OF CONICS. But he was cleurly in error ; for, when the sides are doubled, the area becomes four times as great, and the solid content eight times as great. Geometei-s also continued to investigate the question in wliat manner one miglit double a given solid wliile it remained in the same form. And a problem of this kind was called the doubling of the cul>e ; for they starttnl from a culie and sought to double it. While then for a long time everyone was at a loss, Hippocrates of (Miios was the first to ohser\e that, if between two straight lines of which the greater is double of the less it were discovered how to find two mean proportionals in continueil proportion, the cube would be doubled ; and thus he turned the dilKculty in the original problem* into another difliculty no less than the former. Afterwards, they say, some Delians attempting, in accordance with an oracle, to double one of the alturs fell into the same difficulty. And they sent and liegged the geomettM-s who were with Plato in the Academy to find for them the required solution. And while they set themselves energetically to work and sought to find two means between two given straight lines, Archytas of Tarentum is said to have dis- covered them by means of half-cylinders, and Eudoxus by means of the so-called curved lines. It is, however, characteristic of them all that they indeetl gave demonstrations, but were unable to make the actual construction or to reach the point of practical application, except to a small extent Menaechmus and that with difficulty." Home verses at the end of the letter, in commending Eratosthenes' own solution, suggest that there need be no resort to Archytas' unwieldy contrivances of cylinders or to " cutting the cone in the triiuls of Menaechmus t." This last phrase of Eratosthenes appears Poli/iilus of Euripides, but tlmt the words after σφα\(ΐί (or σφαλϋ^) are Eratosthenes' own, iind that the verses from the trapedy are simply μικρόν y' tXeioi βασιλικού α-ηκον τάφου' διτλάίΤίΟϊ ίστω• τοΰ κύβου δί μΐ) σφα\β^. It would, however, be strange if Eratosthenes had added words merely for the puqjOKe of correetinji them again : and Nauck (Tragicuruvi Graecorum Frnijmenta, Leipzig, ItWJ, p. 871) gives the three verses as above, but holds that they do not belong to the lOlyidus, adding that they are no doubt from an earlier poet than Euripides, perhaps Aeschylus. • TO άκόρημα αύτοΰ is translated by Heiberg " haesitatio eius," which no doubt means " his difliculty." I think it is better to regard αντοΰ as neuter, and as referring to the problem of doubling the cube. + μηδί Mii'^x/ii/oi't κωνοτομΰν τριάδαί. MENAECHMUS. xix again, by way of confirmatory evidence, in a passage of Proclus*, wliere, quoting Geniinus, he says that the conic sections were discovered by Menaechmus. Thus the evidence so far shows (1) that Menaechmus (a pupil of Eudoxus and a conteniporary of Phito) was the discoverer of the conic sections, and (2) that lie used them as a means of solving the problem of the doubling of the cube. We learn fui-ther from Eutociust that IMenaechmus gave two solutions of the problem of the two mean proportionals, to which Hippocrates had reduced the oi-iginal problem, obtaining the two means first by the intersection of a certain parabola and a certain rectangular hyperbola, and secondly by the intersection of two parabolas J. Assuming that a, b are the two given unequal straight lines and .r, y the two required mean proportionals, the discovery of Hippocrates amounted to the discovery of the fact that from the relation !^=i=f (1) X y b it follows that C-Y .- ^ , and, if a - 2b, a? = 2x\ The equations (1) are equivalent to the three equations x^ = ay, y- = bx, xy = ab (2), and the solutions of Menaechmus described by Eutocius amount to the determination of a point as the intersection of the curves represented in a rectangular system of Cartesian coordinates by any two of the equations (2). Let AO, BO be straight lines placed so as to form a right angle at 0, and of length «, b respectively §. Produce BO to χ and AO to y. * Comm. on End. τ., p. Ill (ed. Friedlein). The passage is quoted, witli the context, in the work of Bietschneider, Die Geometrie nnd die Geometer vor Kuklides, p. 177. t Commentary on Archimedex (ed. Heiberg, in. p. 92—98). X It must be borne in mind that the words parabola and hyperbola could not have been used by Menaechmus, as will be seen later on ; but the phraseolofiy is that of Eutocius himself. § One figure has been substituted for the two given by Eutociue, so as to make it serve for both solutions. The figure is identical with that attached to the second solution, with the sole addition of the portion of the rectangular hyperbola used in the first solution. It is a curious circumstance that in Eutocius' second figure the straight line 62 XX THE EAHI.IEH HISTOUY OF COXICS. The firsi solution now consists in drawing a parabola, with vertex Ο and axis Ox, such that its parameter is equal to BO or h, and a hyperhola with Ox, Oy as asymptotes such that the rectangle under the distances of any point on the curve from Ox, Oy respec- tively is equal to the rectangle under Λ0, BO, i.e. to ah. If Ρ be * 1 k- . V (y 0, ^^^^^^ β ο A 1 * the point of intersection of the parabola and hyperbola, and PN, PM be drawn peiju'ndicular to Ox, Oy, i.e. if PN, PM be denoted by y, X, the coordinates of the point P, we shall have y-r^b.ON = b.PM=L• and :cy = PM.PN^ab Ί whence a _x _y X y b' Tn the second solution of Menaechmus we are to drau the parabola descriU'd in the first solution and also the parabola whose rejireeentinR the length of the parameter of each parabola is drawn in the same KtraiRht line with the axiB of the parabola, whereas Apollonius always draws the |pariinn'ter aH a line starting from the vertex (or the end of a diameter) and iw'n)cndicular to the axis (or diameter). It is po.s.'iible that we may have here an additional indication that tlie idea of the parameter as όρθΙα or the /«ii/.s• rectum orij;inat<.'d with Apollonius; thoul•;!! it is also possible that tlie selection of the directions of A(), JU) was due to notliing more than accident, or may have been made in order that the successive terms in the continued proportion minht appear in the figure in cyclic order, which corresponds moreover to their relative positions in the mechanical solution attributed to Plato. For this solu- tion H«'e the same passage of Eutociue (Archimfdfs, ed. Heiberg, in. p. 66 — 70). MENAECHMUS. χχΐ vertex is 0, axis Oy and parameter equal to a. The point Ρ where the two parabohis intersect is given by ar = ay , , . a X y wlience, as before, - = - = !f . X y b We have therefore, in these two solutions, the paralwla and the rectangular hyperbola in the aspect of loci any points of which respectively fulfil the conditions expressed by the equations in (2); and it is more than probable that the discovery of IVlenaochmus was due to efforts to determine loci possessing these characteristic pioperties rather than to any idea of a systematic investigation of the sections of a cone as such. This supposition is confirmed by the very special way in which, as will be seen presently, the conic sections were originally produced from the right circular cone ; indeed the special method is difficult to explain on any other assumption. It is moreover natural to suppose that, after the discovery of the convertibility of the cube-problem into that of finding two mean proportionals, the two forms of the resulting equations would be made the subject of the most minute and searching investigation. The form (1) expressing the equality of three ratios led naturally to the solution attributed to Plato, in which the four lines representing the successive terms of the continued pro- l^ortion are placed mutually at right angles and in cyclic order round a fixed point, and the extremities of the lines are found by means of a rectangular frame, three sides of which are fixed, while the fourth side can move freely parallel to itself. The investigation of the form (2) of the equations led to the attempt of Menaechmus to determine the loci corresponding thereto. It was known that the locus represented by y^ = Χι.τ.,, where y is the perpendicular from any point on a fixed straight line of given length, and x^, x, are the segments into which the line is divided by the perpendicular, wjvs a circle ; and it would be natural to assume that the equation y' = bx, differing from the other only in the fact that a constant is sub- stituted for one of the variable magnitudes, would be capable of representation as a locus or a continuous curve. The only difficulty Avould be to discover its form, and it was here that the cone was introduced. If an explanation is needed of the circumstance that Menaech- XXll THE EAULIKU HISTORY OF CONICS. mus should liavc h.-ul recourse to any solid figure, <and to a cone in piirticulfir, for tlie purpose of producing a plane locus, we find it in the fact that solid geometry had alreivdy reached a high state of development, jus is shown by the solution of the problem of the two mean proportionals by Archytas of Tarentum (born about 430 B.C.). This solution, in itself perhaps more remarkable than any other, determines a certain point as the intersection of three surfaces of revtdution, (1) a right cone, (2) a right cylinder whose base is a circle on the axis of the cone ivs diameter and passing through the ft|)ex of the cone, (3) the surface formed by causing a semicircle, whose diameter is the same as that of the circular base of the cylinder and whose plane is perpendicular to that of the circle, to revolve al)out the apex of the cone as a fixed point so that the diameter of the semicircle nujve.s always in the plane of the circle, in other words, the surface consisting of half a uplit ring whose centre is the apex of (he cone and whose inner diameter is indefinitely small. We find that in the course of the solution («) the intersection of the surfaces (2) and (3) is said to be a certain curve (γραμμην rira), being in fact a curve of double curvature, (h) a circular section of the right cone is used in the proof, and (c), as the penultimate step, two mean proportionals are found in one and the same plane (triangular) section of th.e cone*. • The solution of Archytas is, like the others, given by Eutocius (p, ;»8— 102) nntl is so instructive that I cannot forbear to quote it. Suppose that AC, AB are the strai^'ht hncs between wl)ich two mean proportionals are to be found. AC 18 then made the diameter of a circle, and AD is placed as a chord in the circle. A Bcmicircle is drawn with diameter AC but in a plane perpendicular to that »i AUC, and revolves alwut an axis throuRh .1 perpendicular to the plane of ABC. MENAECHMUS. χχϋί Thus the introduction of cones by Menaechnius should not in itself be a matter for surprise. Concerning JNIenaeclinius' actual method of deducing the proper- ties of the conic sections from the cone we have no definite information ; but we may form some idea of his probable procedure A half-cylinder (right) is now erected with ABC as base: this will cut the surface described by the moving semicircle APC in a certain curve. Lastly let CD, the tanjicnt to the circle ABC at the point C, meet Alt produced in I); and suppose the triangle ACD to revolve about AC as axis. This will generate the surface of a right circular cone, and the point Β will describe a semicircle BQE perpendicular to the plane of ABC and having ita diameter BE at right angles to AC. The surface of the cone will meet in some point Γ the curve described on the cylinder. Let APC be the conesponding position of the revolving semicircle, and let AC meet the circle ABC in M. Drawing PM perpendicular to the plane of ABC, we see that it must meet the circumference of the circle ABC because Ρ is on the cylinder which stands on ABC as base. Let AP meet the circumference of the semicircle BQE in Q, and let AC meet its diameter BE in N. Join PC, QM, QN. Then, since both semicircles are pei^pendicular to the plane ABC, so is their line of intersection QN. Therefore QN is perpendicular to BE. Hence QN-=BN . NE = AN . NM. Therefore the angle AQM is a right angle. But the angle CPA is also right : therefore MQ is parallel to CP. It follows, by similar triangles, that C'A : AP = AP : AM^AM : AQ. That is, AC : AP^AP : AM=AM : AB, and AB, AM, AP, AC are in continued proportion. In the language of analytical geometry, if AC is the axis of x, a line through .1 perpendicular to AC in the plane of ABC the axis of y, and a line through A parallel to PM the axis of z, then Ρ is determined as the intersection of the surfaces x- + U- + '''=^i^' (1). .c--fi/-' = rtx (2), .x- + y- + z'^=ajx'- + y- (3), where AC = a, AB = b. From the first two equations and from this equation and (3) we have a ^ Jx^+y^+z" ^ y/x-'+y' Jx'^ + y-^ + z' V^+I/- l^ or AC:AP=AP:AM=AM:AB. xxiv ΤΗ κ ΚΛΚΙ.ΙΚΚ HISTUUY (»F COMCS. if we bear in mind (1) wljat we are told of the manner in which the earlier writers on conies produced the three curves from particular kinds of rii,dit circular cones, and (2) the course followed by Apol- lonius (and Archimedes) in dealing with sections of any circular cone, whether right or oblique. Eutocius, in his comnientaiy on the Conies of Apollonius, quotes with approval a statement of Geminus to the effect that the ancients defined a cone as the surface described by the revolution of a right- angled triangle about one of the sides containing the right angle, and that they knew no otlier cones than right cones. Of these they dis- tinguishinl three kinds according as the vertical angle of the cone was less than, equal to, or greater than, a right angle. Further they prcKluced only one of the three sections from each kind of cone, always cutting it by a plane perpendicular to one of the generating lines, and calling the respective curves by names corresponding to the particular kind of cone; thus the "section of a right-angled cone " was their name for a parabola, the " section of an acute-angled cone" for an ellipse, and the "section of an obtuse-angled cone" for a hyperbola. The sections are so described by Archimedes. Now clearly the parabola is the one of the three sections for the pnKluction of which the use of a right-angled cone and a section at right angles to a generator gave the readiest means. If iV be a point on the diameter JiC of any circular section in such a cone, and if Λ7' be a straight line drawn in the plane of the section and perpen- dicular to JiC, meeting the circumference of the circle (and therefore the surface of the cone) in J', I'y'-^BN.NC. Draw AM in the plane of the axial triangle OBC meeting the generator OB at right angles in .1, and draw AD parallel to BC meeting OC in D; let DEF, perpendicular to AD or Bt\ meet BC in Ε and AN produced in /'. Then AD is bisected by the axis of the cone, and therefore AF is likewise bisected by it. Draw CG perpendicular to BC meeting A F produced in G. Now the angles Β A iV, BCG are right ; therefore B, A, C, G are (oncyclic, and B.V.NC ^AN.NG. But AN=CD = FG- MENAECHMUS. tlierefore, if .1 F meets the axis of the cone in X, NG = AF^-2AL. Hence PN' = BN.NC ^■2AL.AN, and, if A is fixed, '2AL is constant. ./ 3 η /\J \ L F Ε Thus Ρ satisiies the e(iuati()n rf='2AL..v, where y - PN, χ = A N. Therefore we have only to select A as a point on ()B such that AL (or AO) = ^, and the curve corresponding to the etjuation y^ = bx is found. The 'parameter' of the parabola is equal to twice the distance between A and the point where AN meets the axis of the cone, or ά διπλάσια τα? μίχρι τον άξονος, as Archimedes calls it*. The discovery that the hyperbola represented by the equation xy = ah, where the asymptotes are the coordinate axes, could Ije obtained by cutting an obtuse-angled cone by a plane perpendicular to a generator Λvas not so easy, and it has been (juestioned wliether Menaechmus was aware of the fact. The property, .ry = (const.), for a hyperbola referred to its asympt(jtes does not appear in Apollonius until the second Book, after the diameter-properties haΛ•e been proved. It depends on the propositions (1) that every series of parallel chords is bisected by one and the same diameter, and (2) that the parts of any chord intercepted between the curve and the asymptotes are equal. But it is not necessary to assume that * Cf. On ConoitL• and Spheroids, 3, j). 80 1. XXVI THE EARLIER HISTORY OF CONICS. Mcnapclinius was aware of these general propositions. It is more proljiil.le that he obtained the equation referred to the asymptotes from the equation ref«'rred to tlie axes; and in the particular case which he uses (that of the rectangular hyperbola) this is not difficult. Thus, if /• Ite a point on the curve and J'K, PK' be perpendicular to the iusyniptotes (77ι', CH' of a rectangular hyperbola, and if li'l'XIi' 1m' j>erjK'ndicular to the bisecUir of the angle Vjetween the ii.syniptotes, Ρ Κ . PK' = the rect. CKPK' = the quadrilatei-al CKPE, since aCEK'= APJiA'. Hence PK . FK' ^ A RON - Δ PEN = h{CN^-PN') Nslicrc .'•, // an• the coonlinates of /' referied to the axes of the liyjKTbola. We have then U> sljow iiow MeiKWJchnius could obtain from an obtuse-anglt'd cone, by a section perpendicular to a generator, the H'ctangular hyperlntla a:' - y* ^ (const.) = - , say, 4 or y« _ avr.„ when• ./•,, r, are the distances of the foot of the ordinate y from the jMiints yl, yl' respectively, and Λ A' -a. MENAKCHMrS. χχνϋ Take an obtuse-angled cone, and let BC be the diameter of any circular section of it. Let A be any point on the generator OB, and through A draw AN -Ai right angles to OH meeting CO produced in A' and BC in N. Let y be the length of the straight line drawn from Ν perpen- dicular to the plane of the axial triangle OBC and meeting the surface of the cone. Then y will be determined by the equation f^BN.NC. Let AD be drawn, as before, parallel to BC and meeting OC in D, and let OL, DF, CG be drawn perpendicular to BC meeting AX produced in Z, F, G respectively. Then, since the angles BAG, BGG are right, the points />, A, C, G are coney clic ; .•. y- = BN.NC = AN.NG. But NG : AF= CN : AD, by similar triangles, ^A'N : AA'. AF Hence AN. '^,.Α'Ν AA 2AL - AA'••^'-' and the locus of the extrenuty of y fur different positions of tlie circular section, or (in other words) the section of the cone by a plane through ^xV perpendicular to the plane of the axial triangle, satisfies the desired condition pro cidcl thai -. ., ^^• XXVMl THE EAHLIKH HISToUV oF CONICS. This i-elation, together with the fact that the angle AOL is equal to half the supplement of the angle A'OA, enables us to determine the i)osition of tlie apex (f, and therefore the vertical angle, of the desired cone which is to contain the rectangular hyperbola. For suppose determined, and draw the circle circumscribing AOA' ; this will meet LO produced in some point K, and OA' will l»e its diameter. Thus the angle A'KO is right ; .•. _ Λ A' Κ = complement of .ALK= ^AOL = ^ LOO - _ A'OK, whence it follows that the segments AK, A' Κ are equal, and therefore A' lies on the line })isecting A A' at right angles. Hut, since the angle ^ΓΑ'Λ- is right, A' also lies on the semicircle with A'L as diameti^r. A' is therefore detcniiincd by drawing that semicircle and then drawing a line bisecting A A' at right angles and meeting the semicircle. Thus, A' being found and A' Z» joined, is determined. The foregoing construction for a recttmgular hyperbola can be • •«lii.illy well applied to the case of the hyperbola generally or of an 2.1 Λ fllipse ; only the value of the const;int - -,- will be ditlerent from ' '' AA unity. In every case '2AL is equal to the parameter of the ordinates Ut AA\ or the pai-ameter is equal to twice the distance between the vertex of the section and the axis of the cone, ά διπλάσια tSs μ-ίχρι τον ά^οΐ'ος (as Archimedes called the principal parameter of the parabola). The jissumption that Menaeclinius discovered all three sections in the manner alx)ve set forth agrees with the reference of ICratosthenes to tlie " Menaechmean triads," though it is not im- proliJible that the ellip.se was known earlier as a section of a right cylinder. Thus a passage of Euclid's Phdenomena says, "if a cone or cylinder be cut by a plane not parallel to the base, the resulting section is a section of an acute-angled cone which is similar to a θνρίό%" showing that Euclid distinguished the two ways of pro- ducing an ellipse. Heiberg {Littfrargeschichtliche Studien iiher h'liklid, p. 88) thinks it probable that θνρ^όζ was the name by which Alenaechnms called the curve*. It is a question whether Menaeclimus used mechanical contriv- • The cxpreHeion η τον Ovpeov for the cllipBe occur.s several times in Proclus imd particularly in a passage in which ueminus is quoted (p. Ill) ; and it would seem as though this name for the curve was more common in Geminus' time than the name• "ellipse." [liretschucidcr, p. 170.] MEXAECHMUS. ΧΧΙΧ ances for effecting the coHstruction of his curves. Tlie idea that he did so rests (1) upon the passage in the letter of Eratosthenes* to the effect that all who had solved the problem of the two mean pro- portionals had written theoretically but had not been able to effect the actual consti-uction and reduce the theory to practice except, to a certain extent, Menaechmus and that only with dithculty, (2) upon two well known passages in Plutarch. One of these latter states that Plato blamed Eudoxus, Archytas and Menaechmus for trying to reduce the doubling of the cube to instrumental and mechanical constructions (as though such methods of finding two mean pro- portionals were not legitimate), arguing that the good of geometry was thus lost and destroyed, as it was brought back again to the world of sense instead of soaring upwards and laying hold of those eternal and incorporeal images amid which God is and thus is ever Godt; the other passage {Vita MarceUi 14, § 5) states that, in consequence of this attitude of Plato, mechanics was completely diA-orced from geometry and, after being neglected by philosophers for a long time, became merely a part of the science of war. I do not think it follows from tliese passages that Menaechmus and Archytas made machines for effecting their constructions; such a supposition would in fact seem to be inconsistent Avith the direct statement of Eratosthenes that, with the partial exception of Menaechmus, the three geometers referred to gave theoretical solutions only. The words of Eratosthenes imply that Archytas did not use any mechanical contrivance, and, as regards Menaechmus, they rather suggest such a method as the finding of a large number of points on the curve J. It seems likely therefore that Plato's criticism referred, not to the * See the passage from Eratosthenes, translated above, j). xviii. The Greek of the sentence in question is : συμβέβηκΐ Si ττάσιν αύτοΐί άποδΐίκτικωί ■γ€•γραφ^ι>αι, Xeipovpyrjaai δέ και ets χρΰαν πεσΰν μη δϊψασθαι πλην {πι βραχύ τι του Μίκ^χ/ιοι- καΙ ταΰτα δνσχβρώί. + Διό και Πλάτω;' αι'τό; ίμέμψατο rovs πΐρι Ει'δοξοι» και Άρχύται» και Μ^ναιχμοί' ets opyafiKas και μ-ηχαΐΊκάί KaraaKevas τον τον OTepfoO διπλασιασμύν άττάΊαν έπιχ(ΐρονντα% (ώσττίρ ττΐίρωμένονί δια λόγοι» [scr. δι άλόγοί'] δνο μίσα^ άναΚο-γον μη [scr. η] vapfiKOi λαβΐΐι•). άπόλλί'σθαι γαρ οΰτω καΐ διαφθ(ίρ(σθαι το ■γΐωμ€τρίαί αγαθόν, αϋθΐί ^πΐ τα αισθητά παλινδρομονσηί καΐ μη φΐρομίνη^ άνω, μηδ' άντιΧαμ- βανομένηί των άϊδίων και ασωμάτων ΰκόνων, ττρόί alairtp ών 6 debs del θ(6ί ΐστι. (Quaest. conviv. viii. 2. 1.) Χ This is partly suggested by Eutocius' commentary on Apollonius t. 20, 21, where it is remarked that it was often necessary for want of instruments to describe a conic by a continuous series of points. This passage is quoted by Dr Taylor, Ancient and Modern Geometry of Con/r-s-, p. xxxiii. XXX THE EARLIEK HISTORY OF CONICS. use of machines, but simply to the introduction of mechanical consiilerntioHM in ejvch <jf the three solutions of Archytas, Eudoxus, and Menaechmus. Much hivs been written on the difHculty of reconciling the censure on Archytas and the rest with the fact that a mechanical solution is attril)ute<l by Eutocius to Plato himself. The most proljable explanation is io suppose that Eutocius was mistaken in giving the solution as Plato's ; indeed, h.ul the solution been Plato's, it is scarcely possible that Eratosthenes should not have mentioned it along with the others, seeing that he mentions Plato as having been consulted by tiie Delians on the duplication problem. Zeuthen luus suggested that Plato's objection may have referred, in the case of Menaechmus, to the fact that he was not satisfied to regard a curve as completely defined by a fundamental plane property such as we express by the equation, but must needs give it a geo- metrical definition iis a curve arrived at by cutting a cone, in oi-der to make its f»»rm renli.sable by the senses, though this presentation of it was not m:ule u.se of in the subsequent investigaticms of its pioperties ; but this explanation is not so comprehensible if applied to the objection to Archytas^ solution, where the cui-ve in which the revolving .semicircle and the fixed half-cylinder intei-sect is a curve of double curvature and not a plane curve easily represented by an equation. • CHAPTER II. ARISTAEUS AND EUCLID. We come next to the treatises which Aristaeus ' the elder' and Euclid are said to have written; and it will be convenient to deal with these together, in view of the manner in which the two names are associated in the description of Pappus, who is our authority upon the contents of the works, both of which are lost. The passage of Piippus is in some places obscure and some sentences are put in brackets by Hultsch, but the following represents substantially its effect*. "The four books of Euclid's conies were completed by ApoUonius, who added four more and produced eight books of couics. Aristaeus, who wrote the still extant iive books of nolid loci con- nected with the conies, called one of the conic sections the section of an acute-angled cone, another the section of a right-angled cone and the third the section of an obtuse-angled cone.... ApoUonius says in his third book that the ' locus with respect to three or four lines' had not been completely investigated by Euclid, and in fact neither ApoUonius himself nor any one else could have added in the least to what was written by Euclid with the help of those properties of conies only which had heen proved up to Euclid's time; ApoUonius himself is evidence for this fact when he says that tiie theory of that locus could not be completed without the propositions which he had been obliged to Λvork out for himself. Now Euclid — regard- ing Aristaeus as deserving credit for the discoveries he had already made in conies, and without anticipating him or wishing to construct anew the same system (such was his scrupulous fairness and his exemplary kindline.ss towards all who could advance mathematical science to however small an extent), being moreover in no Λvise con- tentious and, though exact, yet no braggart like the other — wrote so much about the locus as was possible by means of the conies of Aristaeus, without claiming completeness for his demonstrations. ♦ See Pappus (ed. Hultsch), pp. 672— 67β. XXxii THE EAULIEK HISTORY OF COXICS. Had lie done so he would certainly have deserved censure, but, as matters stand, he does not by any means deserve it, seeing that neither is ApoUonius called to account, though he left the most part of liis conies incomplete. ApoUonius, too, has been enabled to add tlir lacking portion of the theory of the locus through having become familiar iK'forehand with what haxl already been written about it by Euclid and having spent a long time with the pupils of Euclid in Alexandria, to which training he owed his scientific habit of mind. Now this ' locus with respect to three and four lines,' the theory of which he is so proud of having added to (though he should rather acknowledge his obligations to the original author of it), is arrived at in this way. If three straight lines be given in position and from one and the same point straight lines be drawn to meet the three straight lines at given angles, and if the ratio of the rectangle contained by two of the straight lines so drawn to the square of the remaining one be given, then the point will lie on a solid locus given in position, that is on one of the three conic sections. And, if straight lines be drawn to meet, at given angles, four straight lines given in position, and the ratio of the rectangle under two of the lines so drawn to the rectangle under the remaining two be given, then in the same way the point will lie on a conic section given in ])Osition." It is necessary at this point to say a word about the solid locus (στίρίό? τόπος). Proclus defines a locus (τόττος) as " a position of a line or a surface involving one and the same property" (γραμμής η ίτΓίφανίίας θίσι<; ποιούσα tv καΐ ταντον σύμπτωμα), and proceeds to say that loci are divided into two classes, line-loci {τόποι προς γραμμαΐς) and siirface-loci (τόποι ττρος ίπίφανβίαις). The former, or loci which are lines, are again divided by Proclus into plane loci and solid loci (τόποι ί'πι'πίδοι and τόποι στ€ρ(οί), the former being simply generated in a plane, like the straight line, the latter from some section of a solid figure, like the cylindrical helix and the conic sections. Similarly Eutocius, after giving as examples of the plane locus (I) the circle which is the locus of all points the perpendiculars from which on a finite straight line are mean proportionals between the segments into which th(; line is divided by the foot of the pcrjM-ndicular, (2) the circle which is the locus of a point whose distances from two fixed points are in a given ratio (a locus investi- gat«*d by ApoUonius in tlu! τόπος ά>'αλιιό/Λ€ΐΌς), proceeds to say that the so-called solid loci have derived their name from the fact that ARISTAEUS ΛΧΠ EUCLID. wxiii they arise from the cutting of solid figures, as for instaiice the sections of the cone and several others*. Pappus makes a fui-ther division of those line-loci which are not i)lane loci, i.e. of the class which Proclus and Eutocius call by the one name of solid loci, into solid loci (στ€ρ€οΙ τόποι) and linear loci (τόττοι γραμμικοί). Thu.s, he says, plane loci may be generally described as those which are straight lines or circles, solid loci as those which are sections of cones, i.e. parabolas or ellipses or hyperbolas, while lineai- loci are lines such as are not straight lines, nor circles, nor any of the said three conic sections t. For example, the curve described on the cylinder in Archytas' solution of the problem of the two mean proportionals is a linear locus (being in fact a curve of double curvature), and such a locus arises out of, or is traced upon, a locus which is a surface (tottos ττρός Ιπιφανύίΐ). Thus linear loci are those which have a more complicated and unnatural origin than straight lines, circles and conies, " being generated from more irregular surfaces and intricate movements;}:." It is now possible to draw certain conclusions from the passage of Pappus above reproduced. 1. The work of Aristaeus on solid loci Λvas concerned with those loci which are parabolas, ellipses, or hyperbolas ; in other words, it was a treatise on conies regarded as loci. 2. This book on solid loci preceded that of Euclid on conies and Λvas, at least in point of originality, more important. Though both treatises dealt with the same subject-matter, the object and the point of view were different ; had they been the same, Euclid could scarcely have refrained, as Pappus says he did, from an attempt to improve upon the earlier treatise. Pappus' meaning must therefore be that, while Euclid wrote on the general theory of conies as Apollonius did, he yet confined himself to those properties which were necessary for the analysis of the solid loci of Aristaeus. 3. Aristaeus used the names "section of a right-angled, acute- angled, and obtuse-angled cone," by which up to the time of Apollonius the three conic sections were known. 4. The three-line and four-line locus must have been, albeit imperfectly, discussed in the treatise of Aristaeus ; and Euclid, in * Apollonius, Vol. ii. p. 184. + Pappus, p. ϋ62. X Pappus, p. 270 : -γραμμαΐ yap ΐτιραι τταρά ras ΰρημίναί d% τ^ιν κατασκΐνην λαμβάνονται ττοικιλωτέραν Ιχοΐ'σαι την yivtaiv και β(βιασμ^νην μάλλοι*, ίξ άτακτο- τέρων (πιφαναων καΐ κινησ€ων tiTi-K(ir\(y μίνων -^ΐννώμΐναι.. Η. C. C xxxiv TMK ΚΛΚΙ.ΙΚΗ HISTORY OF TONICS. dealing syntlieticiilly with tlie same locus, was unable to work out the theory completely because he only used the conies of Aristaeus and did not jxdd fresh discoveries of his own. 5. The Conies of Euclid was superseded by the treatise of ApoUonius, and, though the Solid Loci of Aristaeus was still extant in Pappus' time, it is doubtful whether Euclid's work >vas so. The subject of the three-line and four-line locns will be discussed in some detail in connexion with ApoUonius ; but it may be convenient to mention here that Zeuthen, who devotes some bril- liant chapters to it, conjectures that the imperfection of the investigations of Aristaeus and Euclid arose from the absence of any conception of the hyperbola with two branches as forming one curve (which was the discovery of ApoUonius, as may be in- ferred even from the fulness with which he treats of the double- hyperbola). Thus the proposition that the rectangles under the segments of intei-secting chords in fixed directions are in a constant ratio independent of the position of the point of intersection is proved by ApoUonius for the double-hyperbola as well as for the single branch and for the ellipse and parabola. So far therefore as the theorem was not proved for the double-hyperbola before ApoUo- nius, it was incomplete. On the other hand, had Euclid been in possession of the proof of the theorem in its most general form, then, a.ssuming e.g. that the three-line or four-line locus was reduced by Aristaeus' analysis to this particular property, Euclid would have had the means (which we are told that he had not) of completing the synthesis of the locus also. ApoUonius probably mentions Euclid rather than Aristaeus as having failed to complete the theory for the reason that it Avas Euclid's treatise which was on the same lines as his own ; and, as Euclid was somewhat later in time than Aristaeus, it would in any case be natural for ApoUonius to regard Euclid as the representative of the older and defective investigations which he himself brought to completion. AVith regard to the contents of the Conies of Euclid Λνβ have the following indications. 1. The scope must have been generally the same as that of the first three Books of ApoUonius, though the development of the subject was more .systematic and complete in the later treatise. This we infer from ApoUonius' own preface as well as from the statement of Pappus quoted above. •_'. A more important source of infi>nnalioii, in the sense of ARTSTAEUS A\D EUCLID. XXW giving luore details, is at liand in the works of Archimedes, who frequently refers to propositions in conies as well known and not requiring proof. Thus {(f) Tlie fundamental property of the ellipse, PX' : AN. ΝΛ' = P'N" : AN' . N'A' -- BC" : AC", tliat of tlie hyperbola, PN' -.AN. Ν A' = P'N" : AN' . N'A', and that of the parabola, PN-=p,,.AN, are assumed, and must therefore presumably have been contained in Euclid's work. (b) At the beginning of the treatise on the area of a parabolic segment the following theorems are simply cited. ( 1 ) If Ρ Γ be a diameter of a segment of a parabola and QVq ix chord parallel to the tangent at P, QV = Vq. (2) If the tangent at Q meet VP produced in T, PV= PT. (3) If QVq, Q'V'q' be two chords parallel to the tangent at Ρ and bisected in V, V, PV : PV'^QV : Q'V". '^And these propositions are proved in the elements of conies" (i.e. in Euclid and Aristaeus). (c) The third proposition of the treatise On Conoids and Spheroids begins by enunciating the following theorem : If straight lines drawn from the same point touch any conic section whatever, and if there be also other straight lines drawn in the conic section parallel to the tangents and cutting one another, the rectangles contained by the segments (of the chords) will have to one another the same ratio as the squares of the (parallel) tangents. " And this is proved in the elements of conies ." (d) In the same proposition we find the following property of the parabola : If p„ be the parameter of the ordinates to the axis, and QQ' be any chord not perpendicular to the axis such that the diameter PV bisects it in V, and if QD be drawn perpendicular to PV, then (says Archimedes), supposing ρ to be such a length that QV-.QD'^p:p,, the squares of the ordinates to Ρ Γ (which are parallel to QQ') are equal to the rectangles applied to a straight line equal to ρ and of c'l XXXvi ΤΗΚ EAHLIEll HISTORY OF CONICS. width equal to the respective intercepts on Ρ Γ towards P. "■For thi» has been proved in tJie conies." In otlier words, if /)„, ρ are the parameters corresponding respectively to the axis and the diameter bisecting QQ', P'.p. = QV*:QD\ (For a figure and a proof of this property the reader is referred to the chapter on Archimedes p. liii.) Euclid still used the old names for the three conic sections, but he was aware that an ellipse could be obtained by cutting a cone in any manner by a plane not parallel to the base (assuming the section to lie wholly between the apex of the cone and its base), and also by cutting a cylinder. This is expressly stated in the passage quoted above (p. xxviii) from the Phaenomena. But it is scarcely possible that Euclid had in mind any other than a right cone ; for, had the cone been oblique, the statement would not have been true without a qualification excluding the circular sections subcontrary to the base of tlie cone. Of the contents of Euclid's Surface-loci, or τόποι προ? eVi^avcta, we know nothing, though it is reasonable to suppose that the treatise dealt with such loci as the surfaces of cones, spheres and cylinders, and perhaps other surfaces of the second degree. But Pappus gives two lemmas to the Surface-loci, one of which (the second) is of the highest importance*. This lemma states, and gives a complete proof of, the proposition that the locus of a point whose distance from a given point is in a given ratio to its distcmce from a fixed line is a conic section, and is an ellipse, a parabola, or a hyperbola according as the given ratio is less than, equal to, or greater than, unity. The proof in the case where the given ratio is different from unity is shortly as follows. J^t .S' be the fixed point, and let SX be the perpendicular from aS" on the fixed line. Let Ρ be any point on the locus and PN perpen- dicular to SX, so that SP is to XX in the given ratio. Let e be this ratio, so that '^ ~ NX•' ■ Now let Κ be a point on the line SX such that ~ XK' ' • Pappus (ed. Hultsch) p. Ιϋϋϋ seqq. ARISTAEUS ΛΝΊ) EUCLID. then, if A" be another point so taken that NK = NK\ we shall have• , ΡΙί' + SN' SN' PN' PN' NX' NK' ~ NX' - NK^ ~ XK . XK' The position of the points N, K, K' changes with the position of I'. If we suppose A to be the point on which Ν falls when Κ coincides with Λ', we have SA _ _SN AX'^" NK' KAN SK' A Κ Ν K'S It follows that -^ , „-T^ are both known and equal, and therefore SX SK r, i > TTTr are both known and equal. Hence either of the latter SA ' SN ^ expressions is equal to S X - SK XK SA-SN' "*'' AN' 'hich is therefore known Γ '"^^ 1 Π XXXVin THE EAULli:i{ HISTORY OF CONICS. In like iniiiiner, if A' be the point on which iV falls when K' coincides with Λ', we liave ' , ^. - « ; and in the same way we shall XK' tind that the n-.tio ., „ is known and is equal to A Ν ' Hence, by multiplication, the ratio . ..' , , -τ has a known value. And, since yj-. — ^-, = e', from above, This is the property of a central conic, and the conic will be an ellipse or a hyperbola according as β is less or greater than 1 ; for in the former case the points A, A' will lie on the same side of X and in the latter case on opposite sides of X, while in the former case Ν will lie on A A' and in the latter Ν will lie on A A' produced. The case where e = 1 is easy, and the proof need not be given here. We can scarcely avoid the conclusion that Euclid must have used this pnjposition in the treatise on snrface-loci to which Pappus' lemma refers, and that the necessity for the lemma arose out of the fact that Euclid did not prove it. It must therefore have been assumed by him as evident or quoted as well known. It may therefore well be that it was taken from some known work*, not impossibly that of Aristaeus on solid loci. That Euclid should have been acquainted with the property of conies referred to the focus and directrix cannot but excite surprise It is interesting to note in this connexion another passage in Pappus where he is discussing the various methods of trisecting an angle or circular arc. He gives (p. 284) a method which " some " had used and which involves the construction of a hyperbola whose eccentricity is 2. Suppose it is a segment of a circle which has to be divided into three equal ))arts. Suppose it done, and let .ST be one-third of the arc SPR. Join RP, SP. Then the angle RSP is equal to twice the angle SRP. ARISTAEUS ΛΝΊ) KICLID. XXXIX seeing that this property does not appear at all in Aimllonius, and the focus of a parabola is not even mentioned by him. The ex- planation may be that, as we gather from the preface of Apollonius, he does not profess to give all the properties of cpnics known to him, and his third Book is intended to give the means for the svTitliesis of solid loci, not the actual determination of them. The focal property may therefore have been held to be a more suitable subject for a treatise on solid loci than for a work on conies proper. We must not assume that the focal properties had not, up to the time of Apollonius, received much attention. The contrary is indeed more probable, and this supposition is supported by a remarkable coincidence between Apollonius' method of determining the foci of a central conic and the theorem contained in Pappus' 31st lemma to Euclid's Porisnis. This theorem is as follows : Let Λ'Λ be the diameter of a semi- circle, and from A', A let two straight lines be drawn at right angles to A'A. Let any straight line HH' meet the two perpendiculai-s in R, R' respectively and the semicircle in Y. Further let YS be drawn perpendicular to RR', meeting A'A produced in S. It is to be proved that AS.SA' = AR.A'R', i.e. that SA : AR = A'R' : A'S. Now, since R', A', Y, S are concyclic, the angle A'SR' is equal to the angle A'YR' in the same segment. Let SE bisect the angle RSP, meeting RP in Ε and draw EX, PN perpen- dicular to RS. Then the angle ERS is equal to the angle ESR, so that RE = ES; .•. RX=XS, and X is given. Also RS : SP=RE : EP = RX : XN ; .•. RS -.RX^SP -.NX. But J?.S' = 2i?A'; .•. .ST = 2.VA', whence SP"- = iNX-, or PN- + SN"-=iNX": " Since then the two points .S', A' are given, and PX is perpendicular to SX, while the ratio of NX- to PN- + SN^ is given, Ρ lies on a hyperbola." This is obviously a particular case of the lemma to the τόποι πρόί (ττιφανείμ, Ν Υ - and the ratio „»,^ίΓ77.. 's stated in the same form in both cases. PN- + SN- THE EARLIER HISTORY OF CONICS. Similarly, the angle AJiiS is equal to the angle AYS. But, since A' Υ A , R' YS are both right angles, -A'YR' = ^AYS; .•. ^A'SE'=^ -ARS; hence, by similar triangles, A'R• : A'S = iSA : AR, or AS.SA' = AR.A'R'. It follows of course from this that, if the rectangle AR . A'R' is constant, AS .SA is also constant and -S' is a fixed point. It will be observed that in Apollonius, in. 45 [Prop. 69], the complete circle is used, AR, A'R' are tangents at the extremities of the axis A A' of a conic, and RR' is any other tangent to the conic. Ho has already proved, iii. 42 [Prop. 66], that in this case AR . A' R' - BC*, and he now takes two points S, S' on the axis or the axis produced such that AS . SA' = AS' . S'A' = JiC\ He then proves that RR' subtends a right angle at each of the points .V, θ", and proceeds to deduce other focal properties. Thus Apollonius' procedure is exactly similar to that in the lemma to Euclid's Porisiiis, except that the latter does not bring in the (.•οΐΜΐ•. This fact goes far to support the view of Zeuthen as to the origin and aim of Euclid's Porisms, namely, that tliey were jiartly a sort of by-product in the investigation of conic sections and })artly a means devised for the furtiier development of the subject. CHAPTER III. ARCHIMEDES. No survey of the history of conic sections could be complete without a tolerably exhaustive account of everything bearing on the subject which can be found in the extant works of Archimedes. There is no trustworthy evidence that Archimedes wrote a separate work on conies. The idea that he did so rests upon no more substantial basis than the references to κωνικά στοιχεία (without any mention of the name of the author) in the passages quoted above, which haΛ'e by some been assumed to refer to a treatise by Archi- medes himself. But the assumption is easily seen to be unsafe when the references are compared with a similar reference in another passage* \vhere by the words iv rfj στοίχειωσα the Elements of Euclid are undoubtedly meant. Similarly the words " this is proved in the elements of conies " simply mean that it is found in the text-books on the elementary principles of conies. A positive proof that this is so may be drawn from a passage in Eutocius' commentary on Apollonius, Heracleidest, the biographer of Archi- medes, is there quoted as saying that Archimedes was the first to invent theorems in conies, and that Apollonius, having found that tiiey had not been published by Archimedes, appropriated them J ; * Oh lite Sphere luid Cylinder, i. p. 2i. The proposition quoted is Eucl. xii. '2. t The name appears in the passage referred to as 'RpaKXeios. Apollonius (ed. Heiberg) Vol. ii. p. 168. ί Heracleides' statement that Archimedes was the first to "invent" ((ΐΓίνοησαή theorems in conies is not easy to explain. Bretschneider (p. 156) puts it, as well as the charge of plagiarism levelled at Apollonius, down to the malice with which small minds would probably seek to avenge tiiemsolvos for the contempt in which they would be held by an intellectual giant like xlii THE ΕΛΗΜΚΙί HISTORY OF CONICS. and Eutocius subjoins the remark that the allegation is in his opinion not true, " for on the one hand Archimedes appears in many passages to have referred to the elements of conies as an older treatise (ως παλαιοτίρας), and on the other hand Apollonius does not profess to be giving his own discoveries." Thus Eutocius regarded the refei-ence as being to earlier expositions of the elementary theory of conies by other geometers : otherwise, i.e. if he had thought that Archimedes referred to an earlier work of his own, he would not have used the word παλαιοτέρας but rather some expression like πρότίρον ίκ8(8ομίνης. In searching for the various propositions in conies to be found in Archimedes, it is natural to look, in the first instance, for indica- tions to show how far Archimedes was aware of the possibility of jiroducing tiie three conic sections from cones other than right cones and by plane sections other than those perpendicular to a generator of the cone. We observe, iirst, that he always uses the old names "section of a right-angled cone" «tc. employed by Aristaeus, and there is no doubt that in the three places where the word ίλλειι/^ις appears in the Mss. it has no business there. But, secondly, at the very l)eginning of the treatise On Conoids and Spheroids we find the following : " If a cone be cut by a plane meeting all the sides of the cone, the section will be either a circle or a section of an acute- angled cone" [i.e. an ellipse]. The way in which this proposition was proved in the case where the plane of section is at right angles to the plane of symmetry can be inferred from propositions 7 and 8 of the same treatise, where it is .shown that it is possible to find a cone of wliich a given ellipse is a section and whose apex is on a straight line drawn from the centre of the ellipse (1) perpendicular to the plane of the ellipse, (2) not perpendicular to its plane, but lying in a plane at right angles to it and passing through one of the axes of the ellipse. The problem evidently aniounts to determining the Apollonius. Heiberg, ou the other hantl, thinks that this is unfair to Hera- cleides, who was probably misled into making the charge of plagiarism by finding many of the propositions of Apollonius already quoted by Archimedes as known. Hcibcrg holds also that Heracloides did not intend to ascribe the actual invention of conies to Archimedes, but only meant that the olementary theory of conic sections as formulated by Apollonius was due to Archimedes ; otherwise Eutocius" contradiction would have taken a different form and he would not have omitted to point to the well-known fact that Menaechmus was the dieooverer of the conic sectioue. AKCIIIMKDK: :liii circular sections υ£ the cone, and this is wliat Archiniedcs proceeds to do. (1) Conceive an ellipse with />'/>' as its minor axis and lying in a plane perpendicular to the plane of the paper : suppose tiie line CO drawn perpendicular to the plane of the ellipse, and let be the apex of the required cone. Produce OB, OC, OB', and in the same plane with them draw BED meeting OC, OB' produced in E, D respectively, and in such a direction that BE.ED-.EO^^CA^.CO- (where CA is half the major axis of the ellipse). And this is possible, since BE . ED ■.EO'>BC. CB' : CO-. [Both the construction and this last proposition are assumed as known.] Now conceive a circle with BD as diameter draAvn in a plane perpendicular to that of the paper, and describe a cone passing through this circle and having for its apex. We have then to prove that the given ellipse is a section of this cone, or, if Ρ is any point on the ellipse, that Ρ lies on the surface of the cone. Draw PN perpendicular to BB'. Join OX, and produce it to meet BD in M, and let MQ be drawn in the plane of the circle on BD as diameter and perpendicular to BD, meeting the circumference of the circle in Q. Also draw FG, Η Κ thi-ough Ε, Μ respectively each parallel to BB' . xliv THE EARLIER HISTORY OF CONICS. Now (JM' : //.]f . Μ Κ - η Μ . MD : ΙΠΓ . Μ Κ = BE.ED:FE.EG = {BE . ED : ΕΟη . (ΕΟ' : FE . EG) = {CA"-:CO-).{CO"-:BC .CB') = CA-.BC.CB' ^ΡΝ'•.ΒΝ.ΝΒ•. . . QAP : PiV- = HM. MK : BN . NB' ^ OM' : 0N\ whence, since PN, QM are parallel, OPQ is a straight line. But Q is on the circumference of the circle on BD as diameter ; therefore OQ is a generator of the cone, and therefore Ρ lies on the cone. Thus the cone passes through all points of the given ellipse. (2) Let OC not be perpendicular to AA' , one of the axes of the given ellipse, and let the plane of the paper be that containing -LI' and 0(\ so that the plane of the ellipse is perpendicular to that phme. Ijet BB' l)e the other axis of the ellipse. Now OA, OA' are unequal. Produce OA' to D so that OA .loin AD, and (h-aw FG through C parallel to it. OD. ARCHIMEDES. xlv Conceive a plane tluOUjih AD perpendiculai• to tin• plan»• «.f tlie paper, and in it describe either (it), if CB- - Ft' . CG, a circle with diameter A/J, or (b), if not, an ellipse on AD as axis such that if d he the other axis d'.An'=CJr-:FC.CG. Take a cone with apex and passing through the circle or ellipse just drawn. This is possible even when the curve is an ellipse, because the line from to the middle point of AD is perpen- dicular to the plane of the ellipse, and the construction follows that in the preceding case (1). Let Ρ be any point on the given ellipse, and we have only to ρΓΟΛ'β that Ρ lies on the surface of the cone so described. Draw PX perpendicular to A A'. Join ON, and produce it to meet AD in M. Through Μ draw HK parallel to A' A. Lastly, draw MQ perpendicular to the plane of the paper (and therefore perpen- dicular to both Η Κ and AD) meeting the ellipse or circle about AD (and therefore the surface of the cone) in Q. Then QM' : HM. MK={QM' : DM. MA) . {DM. MA : HM . MK) = {d' : Αΰη . (FC . CG : A'C . CA) = (CB' : FC.CG).{FC.CG : A'C. CA) = CB':A'C.CA = PN^:A'N.NA. .•. QM' : PN^ = HM . MK : ΑΊΥ . NA = 03P : 0N\ Hence OPQ is a straight line, and, Q being on the surface of the cone, it follows that Ρ is also on the surface of the cone. The proof that the three conies can be produced by means of sections of any circular cone, whether right or oblique, which are made by planes perpendicular to the plane of symmetry, but not necessarily perpendicular to a generating line of the cone, is of course essentially the same as the proof for the ellipse. It is therefore to be inferred that Archimedes was equally aware of the fact that the parabola and the hyperbola could be found otherwise than by the old method. The continued use of the old names of the curves is of no importance in this connexion because the ellipse was still called the "section of an acute-angled cone" after it was discovered that xlvi THE KARLIF.H HISTORY OF CONICS. it could ])e pnjducwl by means of a plane cutting all the generating lines of any cone, whatever its vertical angle. Heiberg concludes that Archimedes only obtained the parabola in the old way because he describes the parameter as double of the line betAveen the vertex of the paralx)la and the axis of the cone, which is only correct in the case of the right-angled cone ; but this is no more an objection to the continued use of the term as a well-known description of the parameter than it is an objection to the con- tinued use by Archimedes of the term "section of an acute-angled cone" that the ellipse had been found to be obtainable in a different manner. Zeuthen points out, as further evidence, the fact that we have the following propositions enunciated by Archimedes Λvithout pioof {On Conoids and Spheroids, 11) : (1) "If a right-angled conoid [a paraboloid of revolution] be cut by a plane through the axis or parallel to the axis, the section will be a section of a right-angled cone the same as that compre- hending the figure (ά αντά τα ττεριλαμβαΐΌνσα το σχήμα). And its diameter [axis] will be the common section of the plane which cuts the figure and of that which is draΛvn through the axis perpen- dicular to the cutting plane. (2) " If an obtuse-angled conoid [a hyperboloid of revolution] be cut by a plane through the axis or parallel to the axis or through the apex of the cone enveloping (πΐρύχοντυς) the conoid, the section will ])e a section of an obtuse-angled cone : if [the cutting plane passes] through the axis, the same as that comprehending the figure: if parallel to the axis, similar to it : and if through the apex of the cone enveloping the conoid, not similar. And the diameter [axis] of the section will be the common section of the plane which cuts the figure and of that drawn through the axis at right angles to the cutting plane. (3) " If any one of the spheroidal figures be cut by a plane through the axis or parallel to the axis, the .section will be a section of an acute-angled cone : if through the axis, the actual section which comprehends the figure : if paralle%o the axis, similar to it." Archiniodes adds that the proofs of all these propositions are ob\ious. it is therefore tolerably certain that they were based on the same essential principles as his earlier proofs relating to the .sections of conical surfaces and the proofs given in his later investi- gations of the elliptic sections of the various surfaces of revolution. These depend, as will be seen, on the proposition that, if two chords ARCHIMKDES. drawn in fixed directions intersect in !i point, the ratio of the rect- angles under the segments is independent of the position of the point. This corresponds exactly to the use, in the above proofs with regard to the cone, of the proposition that, if straight lines Fd, IIK are diawn in fixed directions between two lines forming an angle, and if FG, Η Κ meet in any point M, the ratio FM . MG : HM .MK is constant ; the latter property being in fact the particular case of the former where the conic reduces to two straight lines. Tlie following is a reproduction, given by Avay of example, of the proposition (13) of the treatise On Conouh and Spheroids which proves that the section of an obtuse-angled conoid [a hyperboloid of re- volution] by any plane which meets all the generators of the en- veloping cone, and is not perpendicular to the axis, is an ellipse whose major axis is the part intercepted within the hyperboloid of the line of intersection of the cutting plane and the plane through the axis perpendicular to it. Suppose the plane of the paper to be this latter piano, and the line EC to be its intersection with the plane of section which is perpendicular to the plane of the paper. Let Q be any point on the section f»f the hyperboloid, and draw QM perpendicular to liC. xlviii THE EARLIEI? HISTORY OF CONK'S. Lt't ^^ΙΖ-^Ιη' the hyperlxtlic section of the hyperboloid made by the phine of the paper and AD its axis. Through J/ in this plane (h-aw J'JDF at right angles to A J) meeting the hyperbola in E, F. Then the section of the hyperljoloid by the plane through EF perpendicular to AD is a circle, QM lies in its plane, and (? is a point on it. Therefore QM' = EM . MF. Now let PT be that tangent to the hyperbola Avhich is parallel to BC\ and let it meet the axis in Τ and the tangent at A in 0. Draw /'Λ' perpendicular to AD. Then QM- : BM . MC = EM . MF : Β Μ . MC = OA' : OP'; which is constant for all positions of Q on the section through BC. Also OA < OP, because it is a property of hyperholas that AT<AN, and therefore OT<OP, whence a fortiori OA <0P. Therefore Q lies on an ellipse whose major axis is BC. It is also at once evident that all parallel elliptic sections are similar. Archimedes, it will be seen, here assumes two propositions (rt) that the ratio of the rectangles under the segments of intersecting chords in fixed directions is equal to the constant ratio of the squares on the parallel tangents to the conic, and (0) that in a hyperbola AN>AT. The first of these two propositions has already been referred to as liaving been known before Archimedes' time [p. xxxv] ; the second assumption is also interesting. It is not easy to see how the latter could be readily proved except by means of the general property that, if PP' be a diameter of a hyperbola and from any point Q on the curve the ordinate QV be drawn to the diameter, while the tangent QT meets the diameter in 2\ then rP : TP' = PV : P' V, so that we may probably assume that Archimedes was aware of this property of the liyperbola, or at least of the particular case of it where the diameter is the axis. It is certain that the corresponding general proposition for the paralxila, PV ΡΊ\ was familiar to him ; for he makes frequent use ..f it. ARCHIMEDES. χΗχ As a preliminary to collecting and arranging in order the otlici• properties of conies either assumed or proved by Archimedes, it may be useful to note some peculiarities in his nomenclature as compared with that of Apollonius. The term diameter, when used with reference to the complete conic as distinguished from a segment, is only applied to what was afterwards called the axis. In an ellipse tlie major axis is ά μάζων Sta/xcrpo? and the minor axis a. «λασσων 8ιάμ(τροζ. For the hyperbola, by the ' diameter ' is only understood that part of it which is within the (.single-branch) hj^erbola. Tiiis Λνβ infer from the fact that the ' diameter ' of a hyperbola is identified with the axis of the figiire described by its revolution about the diameter, while the axis of the hyperboloid does not extend outside it, as it meets {άπτεται) the surface in the vertex (κορνφά), and the distance between the vertex and the apex of the enveloping cone [the centre of the revolving hyperbola] is * the line adjacent to the axis ' (d 7Γοτ€ον'σα τω αξονι). In the parabola diameters other than the axis are called * the lines parallel to the diameter ' ; but in a segment of a parabola that one which bisects the base of the segment is called the diameter of the segment (τον τμάματος). In the ellipse diameters otlier than the axes have no special name, but are simply ' lines drawn through the centre.' The term axis is only used with reference to the solids of revolution. For the complete figure it is the axis of revolution ; for a segment cut oflf by a plane it is the portion intercepted within tlie segment of the line, (1) in the paraboloid, draΛvn through the vertex of the segment parallel to the axis of revolution, (2) in the hyper- boloid, joining the vertex of the segment and the apex of the enveloping cone, (3) in the spheroid, joining the vertices of the two segments into Avhich the figure is divided, the vertex of any segment being the point of contact of the tangent plane parallel to the base. In a spheroid the ' diameter ' has a special signification, meaning the straight line draΛvn through the centre (defined as the middle point of the axis) at right angles to the axis. Thus we are told that "those spheroidal figures are called similar whose axes have the same ratio to the diameters*." The two diameters (axes) of an ellipse are called conjugate {σνζνγίΐ<;). The asymptotes of a hyperbola are in Archimedes the straight lilies nearest to the section of the obtuse-angled conp (at Ιγγιστα * On Conoidn mid Spheroids, p. 282. H. C. d 1 THE EARLIER HISTORY OF CONICS. eieHaL τας τονί άμβλνγωνίου κώνου το/χα?), while what we call the centre of a liyperbola is for Archimedes the jwint in which the nearest lines meet (to σαμάον, καθ' ο αί εγγιστα συ/χ,τΓίτττοντι). Archimedes never speaks of the * centre ' of a hyperbola : indeed the use of it implies the conception of the two branches of a hyperbola as forming one curve, which does not appear earlier than in Apollonius. When the asymptotes of a hyperbola revolve with the curve round the axis they generate the cone enveloping or comprehenditig the liyperboloid, (τον δί κώνον τον π(.ριΚαφθΙντα νττο ταν £γγιστα τα5 τον αμβλχτγωνίον κώνου To/i.a5 ττίριίχοντα το κωνοειδί? κοΧίίσθαι). The following enumeration* gives the principal properties of conies mentioned or proved in Archimedes. It will be convenient to divide them into classes, taking first those propositions which are either quoted as having been proved by earlier writers, or assumed as known. They fall naturally under four heads. I. General. 1. The proposition about the rectangles under the segments of intersecting chords has been already mentioned (p. xxxv and xlviii). 2. Similar conies. The criteria of similarity in the case of central conies and of segments of conies are practically the same as tliose given by Apollonius. The proposition that all parabolas are similar was evidently familiar to Archimedes, and is in fact involved in his statement that all paraboloids of revolution are sim'ilar (τα μίν ovv ορθογώνια κωνυαΒία πάντα o/ixotci €vti). 3. Tangents at the extremities of a 'diameter' (axis) are perpendicular to it. II. TuE Ellipse. 1. The relations Λν^ : AiV. A'N= FN'- : AN' . A'N' = BB'- :AA" or CB' : CA' * A word of acknowledgement is due here to Heiberg for tlie valuable summary of " Die Kenntnisse des Arcliimedes iiber die Kegelschuitte," contained in the ZeilHchri/l fur Mathematik xtnd Physik {Hintorisch-Iiterarische Abthcihnig) IfiHO, j<p. 41 — Γ)7. This article ie a complete guide to the relevant passages in Arcliimedes, though I have of course not considered myself excused in any instiincf fron> referring to the original. ARCHIMEDES. Η are constantly used as expressing the fundamental property and the criterion by which it is established that a curve is an ellipse. 2. The more general proposition QV -.FV.rV^Q'V" ■.ΡΓ.ΓΎ' also occurs. 3. If a circle be described on the major axis as diameter, and an ordinate PN to the axis of the ellipse be produced to meet the circle in p, then pN : P^== (const.). 4. The straight line drawn from the centre to the point of contact of a tangent bisects all chords parallel to the tangent. 5. The straight line joining the points of contact of parallel tangents passes through the centre ; and, if a line be drawn through the centre parallel to either tangent and meeting the ellipse in two points, the parallels through those points to the chord of contact of the original parallel tangents will touch the ellipse. 6. If a cone be cut by a plane meeting all the generators, the section is either a circle or an ellipse. Also, if a cylinder be cut by tAvo parallel planes each meeting all the generators, the sections will be either circles or ellipses equal and similar to one another. III. The Hyperbola. 1. We find, as fundamental properties, the following, PN^ : P'N" = AN. A' Ν : AN' . A'N\ QV: Q'V" = PV.P'V:PV'.P'r; but Archimedes does not give any expression for the constant ratios PN' : AN. A' Ν and QV^ : PV . P'V, from which we may infer that he had no conception of diameters or radii of a hyperbola not meeting the curve. If Che the point of concourse of the asymptotes. A' is arrived at by producing AC and measuring CA' along it equal to CA ; and the san>e procedure is used for finding /*', the other extremity of the diameter through Ρ : the lengths A A', PP' are then in each case double of the line adjacent to the axis [in one case of the whole surface, and in the other of a segment of which Ρ is tlie 'vertex']. This term for AA', PP' was, no doubt, only used in order to avoid mention of the cone of (12 Hi THE EARLIER HISTORY OV CONICS. which the hyperbola is a section, as the introduction of this cone might have complicated matters (seeing that the enveloping cone also appears); for it is obvious that A A' appeared first as the distance along the principal diameter of the hyperbola intercepted between the vertex and the point where it meets the surface of the opposite half of the double cone, and the notion of the asymptotes came later in the order of things. 2. If from a point on a hyperbola two straight lines are drawn in any directions to meet the asymptotes, and from another point two other straight lines are similarly drawn parallel respectively to the former, the rectangles contained by each pair will be equal*. 3. A line through the point of concourse of the asymptotes and the point of contact of any tangent bisects all chords parallel to the tangent. 4. If PX, the principal ordinate from P, and P2\ the tangent at P, meet the axis in N, Τ respectively, then AN>AT. 5. If a line between the asymptotes meets a hyperbola and is bisected at the point of concourse, it will touch the hyperbola f. IV. The Parabola. 1. PN' :P'N'*=:AN :AN' \ and QV':Q'V" = PV.PV' ]' We find also the forms ΡΝ'^2^α•^Νχ QV'=2y.Pr Γ j)„ (the principal parameter) is called by Archimedes the parameter of the ordinates (parallel to the tangent at the Λ -ertex), τταρ* αν δύνανται αϊ άττυ τα5 το/ιας, and is also described as the do7ible of the line extending [from the vertex] to the axis [of the cone] ά διπλάσια tSs μίχρι τον ΰ^οΐΌζ. The term 'parameter' is not applied by Archimedes to p, the constant in the last of the four equations just given, ρ is simply described as the line to which the rectangle equal to QV- and of width equal to Ρ F is applied. 2. Parallel chords are bisected by one line parallel to tlie axis ; • This proposition aud its converse appear in a fragment given by Eutocius in his note on the 4th proposition of Book ii. On the Sphere and Cylinder. t Tliis is also used in the fragment quoted by Eutocius. ARCHIMEDES. aiul a line parallel tu the axis bisects chords parallel to the tangent at the point where the said line cuts the parabola. 3. If QD be drawn perpendicular to the diameter PV bisecting the chord Q VQ', and \i ρ be the parameter of the ordinates parallel to QQ' , while y^„ is the principal parameter, p:p,, = QV'-:QD\ [This proposition has already been mentioned above (p. xxxv, xxxvi). It is easily derived from ApoUonius' proposi- tion I. 49 [Prop. -22]. li PV meet the tangent at A in E, and PT, A Ε intersect in 0, the proposition in question proves that and OP '.PE = p: 2P1\ OP = },PT ; .•. ΡΓ=^ρ.ΡΕ = p.AN. Thus Q Γ' : QD- = PT' : PN-, by similar triangles, =^ p. AN : Pa. AN = P 'Pa-] ■t. If the tangent at Q meet the diameter Ρ V in Γ, and QV he an ordinate to the diameter, PV=PT. δ. By the aid of the preceding, tangents can be drawn to a parabola («) from a point on it, (ό) parallel to a given chord. 6. In the treatise On floatimj bodies (ττερί tQv οχονμίνων), ii. 5, we have this proposition : If Κ be a point on the axis, and KF be measured along the axis away from the vertex and equal to half the principal parameter, while KII is draΛvn perpendicular to the diameter through any point P, then FH is perpendicular to the tangent at P. (See the next figure.) It is obvious that this is equivalent to the proposition that the subnormal at an// jjoint Ρ is const(tnt (uul equal to half the priii<;iji<d parameter. liv THE EARLIER HISTORY oF CONICS. 7. If QAQ' be a segment of a paraljola such that QQ' is perpendicuhir to the axis, while QV<], parallel to the tangent at P, meets the diameter through Ρ in Γ, and if li be any other point on the curve the ordinate from which RlIK meets PV in // and the axis in /Γ, then (J/ being the middle point of QQ') PV : PlI ^^^MK : Κ A, "/o7• this is proved." {On floating bodies, II. G.) [There is nothing to show where or by Λvhom the proposition was demon- strated, but the proof can be supplied as follows : , PV MK . We have to prove that ^ - is jwsittve or zero. Let Qq meet AM in 0. PV_ Μ Κ _ PV.AK -Ρ Η .MK ~ ΚΑ ~ Now PH PH. Κ A AK . PV - {AK - AN) {A Μ -AK) AK.PH Τ AK'-AK{AM + AN- ΡΓ)4- AM. AN ^ AK.PH AK'-AK. OM+AM. AN AK.PU ' Γ (HxncG AN = AT). OM NT β^*^ QM-PN^ OM* iAN' " p„.AM 2->„.AN' whence OM'=iAM.AN, ΛΜ.ΑΝΛψ. »« It follows that AK* -AK.O.M Λ AM.AX^ ΑΓ--ΑΚ . OM + OM' > i AllCHlMKDKS. which is a complete square, and therefore cannot b(i negative ; 'TV MK\ Iv whence the proposition follows.] 8. If any three similar and similarly situated paraljolic seg- ments have one extremity (β) of their bases common and their bases BQ , BQ.,, BQ.^ lying along the same straight line, and if EO he draΛvn parallel to the axis of any of the segments meeting the tangent at Β to one of them in E, the common base in 0, and each of the three segments in B^, B^, R^, then Ββ^ bq^-q^q: [This proposition is given in this place because it is assumed without proof {On floating bodies, il. 10). But it may well be that it is assumed, not because it was too well known to need proof, but as being an easy deduction from another proposition proved in the Quadrature of a jiarabola which the reader could work out for himself. The latter proposition is given below (No. 1 of the next group) and demonstrates that, if BB be the tangent at Β to the segment BB^(J^ , ER^ : R/J = BO : OQ^. To deduce from this the property enunciated above, we observe first that, if V ^, V^, V^ be tiie middle points of the bases of the three Ivi THE EARLIER IIIS'IORV OF CONICS. segments and the (parallel) diameters through F,, V^, F^ meet the respective segments in Γ^, J\, P^, then, since the segments are simihar, /n\ : B]\ : Ji]\ - I\V, : PJ\ : 1\V.,. It follows that />, 1\, P^, 7^3 are in one straight line. But, since BE is the tangent at Β to the segment BR^Q^, TJ\ = PJ^ (where Γ,Ρ, meets BE in Ί\). Therelforo, if Υ,Ρ,, ]\P^ meet BE in 7;, 7',, V. = ^.''- and ^Λ = ^.η, and />/i' is therefore a tangent to all three segments. Next, since ER^ : Rfi - BO : (?(?,, ER^ : ^0 = 7iO : BQ^ . Similarly ER, : EO = BO : BQ„, and ER^ : EO = BO : 7?^^. From the tirst two relations we derive EO \BQ^ BqJ ^BO.Q.Q, bq.-bq: Similarly R& ^BOJQ^^ .-similarly ^^ BQ^.BQ^ From the last two results it follows that R^r BQ.'QM' 9. If two similar parabolic segments with bases BQ , BQ_, be placed as dt-scribed in the preceding proposition, and if BRJi, be any I' f AllC'llIMKDES. IvU straight line tlirough J> cutting the segments in A',, A', re.si»ectively, then BQ^ : BQ,,- nn^ : Bli^. [Let the diameter through /?, meet the tangent at Β in E, the other segment in A, and the common Ijase in 0. Tlien, as in the last proposition, EB^ : EO = BO : BQ^, and ER.EO^BO: BQ.^ ; .•. ER -.ER^^BQ^ : BQ.,. But, since A, is a point within the segment BR(J,, and A'AA^ is the diameter through A, , we have in like manner ER : ER^ - ^A, : BR^. Hence BQ^ : BQ, = BR^ : BR.^.] 10. Archimedes assumes the solution of the problem of placing, between two parabolic segments, similar and similarly situated as in the last case, a sti'aight line of a given length and in a direction parallel to the diameters of either parabola. [Let the given length be I, and assume the problem solved, A7i, being equal to l. Using the last figure, we have BO ER^ BQ^~ EO' BO ER '""^ bcCeo• Subtracting, we obtain BO.Q ^Q, ^ RR, . BQ, . BQ, EO ' whence /?(9. 0^ - / . ^^^"^^S which is known. And the ratio BO : OE is given. Tiierefore B0\ or OE', can be found, and therefore 0. Lastly, the diameter through determines A A,.] It remains to describe the investigations in which it is either expressed or implied that they represent new developments of the theory of conies due to Archimedes himself. With the exception of ΠΝ1 V Iviii THE ΕΛΚΜΚΙΙ HISTOKV OF COXICS. certain propositions relating to the areas of ellipses, his discoveries mostly have reference to the parabola and, in particular, to the determination of the area of any parabolic segment. The preface to the treatise on that subject (which was called by Archimedes, not Τ€τρα•γωνισμ6<; τταραβοΧη^, but ircpi της τον ορθογωνίου κώνου τομής) is interesting. After alluding to the attempts of the earlier geometers to square the circle and a segment of a circle, he proceeds : " And after thfit they endeavoured to square the area bounded by the section of the Λvhole cone* and a straight line, assuming lemmas not easily conceded, so that it was recognised by most people that the problem was not solved. But I am not aware that any one of my predecessors has attempted to square the .segment bounded by a straight line and a section of a right-angled cone, of which problem I have now discovered the solution. For it is here shown that every segment bounded by a straight line and a section of a right-angled cone is four-thirds of the triangle which has the same base and an equal altitude with the segment, and for the demonstration of this fact the following lemma is assumed f : that the excess by which the greater of (two) unequal areas exceeds the less can, by being added to itself, be made to exceed any given finite area. The earlier geometers have also used this lemma ; for it is by the use of this same lemma that they have shown that circles are to one another in the duplicate ratio of their diameters, and that spheres are to one another in the triplicate ratio of their diameters, and further that every pyramid is one third part of the prism having the same base with the pyramid and equal altitude : also, that every cone is one third part of the cylinder having the same base as the cone and equal altitude they proved by assuming a certain lemma similar to that aforesaid. And, in the result, each of the aforesaid theorems has been accepted ;}: no less than those proved * There seems to be some corruption here : the expression in the text is ras δλου τον κώνου τομάί, and it is not easy to give a natural and intelligible meaning to it. The section of ' the whole cone ' might perhaps mean a section cutting right through it, i.e. an ellipse, and the ' straight line ' might be an axis or a diameter. But Heiberg objects to the suggestion to read tSj όξι^γωνίου κώνου τομαί, in view of tlie addition of /tot ii'^iiay, on the ground that the former expression always signifies the whole of an ellipse, never a segment of it (Qtuiestioties Archiviedeae, p. 1411). t Tiie lemma is used in tlie mechanical proof only (Prop. 16 of the treatise) and not in the geometrical proof, which depends on Eucl. x. 1 (see p. Ixi, Ixiii). ^ The Greek of this passage is : σνμβαΐνΐΐ δί των ττροειρημένων θίωρημάτων AUCHIMKDKS. lix without tlie lemma. As therefore my work now pulilishi'd has satisfied the same test as the propositions referred to, I have written out the proof of it and send it to you, first as investigated by means of meclianics and next also as demonstrated by geometry. Prefixed are, also, the elementary propositions in conies which are of service in the proof " (στοιχεία κωνικά χρ^ΐαν Ι;^οντα es τα^ άπό^ίίξιν). The first three propositions are simple ones merely stated without proof. The remainder, Avhich are given below, were apparently not considered as forming part of the elementary theory of conies ; and this fact, together Avith the circumstance that they appear only as subsidiary to the determination of the areas of parabolic segments, no doubt accounts for what might at first seem strange, viz. that they do not appear in the Conies of Apollonius. 1. 1/ Qq be the base of any segment of a parabola, and Ρ the vertex* of the segment, and if the diameter through any other point R on the curve meet Qq in 0, QP in F, and the tangent at Q in E, then (1) QV.VO = OF:FR, (2) QO •.Oq = FP:POf. (ίκαστον μηδέν ησσον τύν avev τούτου τον λήμματος άποδ€δ€ΐ•γμ4ι>ωΐ' πειτιστευκίναι. Here it would seem that πεπιστ^νκέναι must be wrong and that the Passive should have been used. * According to Archimedes' definition the height (ΰψο%) of the segment is " the greatest perpendicular from the curve upon the base," and the vertex (κορυφά) "the point (on the curve) from which the greatest perpendicular is drawn." The vertex is therefore P, the extremity of the diameter bisecting Qq. t These results are used in the mcchanicnl investigation of the area of a parabolic segment. The mechanical proof is here omitted both because it is more lengthy and because for the present purpose the geometrical proof given below is more germane. Ix THE EAIILIKR HIsniHV OF CONICS. To prove (1), we draw the onliuate 7i' II' to I'V, meeting QP in K. Now J'V : DV^QV : JiW; therefore, by jjaralleLs, PQ : PK=PQ' :PF\ In other words, PQ, PF, PK are in continued proportion ; .•. PQ : PF-^ PF '. Ρ Κ = PF + PQ : Ρ Κ + PF = QF:KF; tlierefore, by parallels, QV : VO^OF : FR. To piOve (2), we obtain from the relation just proved QV : qO = OF • OR. Also, since TP = PV, EF=^ OF. Accordingly, doubling the antecedents in the proportion, Qq:qO^OE: OR, or QO .Oq^ER: RO. It is clear that the equation (1) above is equivalent to a change of axes of coordinates from the tangent and diameter to the chord Qq (as axis of .'.;, say) and the diameter through Q (as the axis of y). d' For, if QV=a, PV nd if QO = X, RO = y, ,-e have at once from (1) _ «_ _ OF . X — a OF - y ' a OF ^' ρ " •2α-χ~ y ~ y ' whence j/y = χ (2fi — x). Zcutlieu points out (p. Gl) that the results (1) and (2) above can be put in the forms RO.OV = FR.qO (1) and RO.OQ^ER.qO (2) ARCHIMEDES. Ixi and either of these equations represents a particular case of the parabola as a "locus with respect to four lines." Thus the first represents the equality of the rectangles formed, two and two, from the distances of the movable point Λ' taken in fixed directions from the fixed lines Qq, PV, PQ and Gq (where Gq is the diameter through q) ; while the second represents the same property with respect to the lines Qq, QD (the diameter through Q), QT ami Gq. 2. If RM he a dianiPter bisectiny QV in J/, and RW be the ordinate to PV from R, then PV = ^RM. For PV :PW=QV' -.RW = ^RW' : RW; .•. PV=iPW, and PV=^RM. 3. The triangle PQq is greater than half the segment PQq. For the triangle PQq is equal to half the parallelogram contained by Qq, the tangent at P, and the diameters through Q, q. It is therefore greater than half the segment. Cor. It follows that a j^olygon can he inscribed in the segment such that the remaining segments are together Jess than any assignable area. For, if we continually take away an area greater than the half, we can clearly, by continually diminishing the remainders, make them, at some time, together less than any given area (Eucl. x. 1). 4. With the same assntyiptions as in No. 2 aboi'e, the triangle PQq is equal to eight times the triangle RPQ. RM bisects Q V, and therefore it bisects PQ (in Y, say). Therefore the tangent at R is parallel to ΐχκ Now PV=^RM, and PV=2Y3f•, .•. yM=2RY, and APQM=2l^PRQ. Hence APQV=iAPRQ, so that APQq = 8APRQ. THE EARLIER HISTORY OF CONICS. Also, if liW produced moot the curve again in r, Δ PQq = 8 Δ Prq, similarly. 5. 1/ there be a sei'ies of areas A, B, C, D... each of which is four times the next in order, and if the largest, A, is equal to the triatigle PQq, then tJie snm of all the areas A, B, C, D... will be less than the area of the parabolic segment PQq. For, since A PQq :^ 8 A PQR = 8 Δ Pqr, Δ PQq = i(APQR + A Pqr) j therefore, since Δ PQq = A, A PQR + APqr = B. In like manner we can prove that the triangles similarly in- scribed in the remaining segments are together equal to the area C, and so on. 1^ Therefore A + B + C + J) + is equal to the area of a certain inscribed polygon, and therefore less than the area of the segment. 6. Given the series A, B, C, D...just described, if Ζ be tlie last of the seft'ies, then A + B + C + ...+z+\z=yA. A Β C ARCHIMEDES. Let d - ^D, and so on, Then, since b = \B, and B+b = },A. Similarly Β + C C + c= IB, Therefore -rD + ...-\-Z+h^-C + d + -h {A + B + C + D+ ... + Y). But b + c + d+. ..+y= 1 (7? + C + Z>+...+ .•. B+C + I)+ ...+Z+Z = }rA, or A + B + C + D+ ... + Z+}^Z=f^A. 7. Every segment bounded by a pcwabola and a chord is four-thirds of the triangle tvhich has the same base and equal altitude. Let K=^.APQq, and we have then to prove that the segment is equal to A". Now, if the segment is not equal to K, it must be either greater or less. Fh-st, suppose it greater. Then, continuing the construction indicated in No. 4, we shall finally have segments remaining whose sum is less than the area by which the segment PQq exceeds Κ [No. 3, Cor.]. Therefore the polygon must exceed Λ' : which is impossible, for, by the last proposition, A+B + C+ ... +Z<*A, where yl = Δ FQq. Secoyidly, suppose the segment less than K. If Δ PQq = A, B--^\A, C^\B, and so on, until we arrive at an area X such that X is less than the difference between Κ and the segment, A+B + C r ... + X + \Χ^^Λ = K. Ixiv THE EARLIER HISTORY OF COXICS. Now, since Κ exceeds A ^ Β λ -C λ- ... ^ X by an area less than X, and the segment l)y an area greater than X, it follows that yl+j5 + C+...+X is greater tlian the segment : which is impossible, by No. 4 above. Tims, since the segment is neither greater nor less than /i", it follows that the segment = A' = ^ , δ PQq. 8. The second proposition of the second Book of the treatise On thr equilibrium of plaries {ίτηπίΒων Ισορροπιών) gives a special term for the construction of a polygon in a parabolic segment after the manner indicated in Nos. 2, 4 and 5 above, and enunciates certain theorems connected with it, in the following passage : " If in a segment bounded by a straight line and a section of a light-angled cone a triangle be inscribed having the same base as the segment and equal altitude, if again triangles be inscribed in the remaining segments having the same bases as those segments and equal altitude, and if in the remaining segments triangles be continually inscribed in the same manner, let the figure so produced be said to be inscribed in the recognised manner {-γνωρίμως ίγγράφίσθαι) in the segment. Atul it is plain (1) that the lines joining the two angles of the figwe so inscribed which are nearest to the vertex of the segment, and the next pairs of angles in order, υήΙΙ be jxirallel to the base of the segment, (2) that the said lines tvill be bisected by the diameter of the segment, and (3) that they will cut the diameter in the proportiojis of (he successive odd numbers, the number one having reference to [the length adjacent ίο] the vertex of the segment. And these properties λυΙΙΙ have to be proved in their proper places (ev ταΓς τα^ίσιν)." These propositions were no doubt established ])y Archimedes by means of the above-mentioned properties of parabolic segments ; and the last words indicate an intention to collect the propositions in systematic order with proofs. But tiie intention does not appear to liave been carried out, or at least Ave know of no lost work of Archimedes in whicli they could have been included. Eutocius proves them by means of Apollonius' Conies, as he does not appear to have seen the work on the area of a parabolic segment ; but the lirst two are easily derived from No. 2 above (p. l.\i). ARCHIMEDES. Ixv The third may be proved as folloAvs. If QiQjQoQ^PQ^Qofl/ly ί»β a- figure -γνωρίμως ΐγγΐ-γραμμ^νον, we lia%e, since <?,<?,, Qj/., ■■■ are all parallel and bisected by /'K, , PI', : PV^ : PV.^ : ΡΓ. ... = 1 : 4 : 9 : 16 ; whence it follows tliat PF, : VV,^ : Γ,Γ^, : V.J\... = 1:3:5:7 .... 9• -(/' QQ' be a chord of a ^>«ί•((όο?α bisected in V by the diameter Ρ V, and if PV is of constant length, then the areas of tL• triangle PQQ' and of the segvtent PQQ' are both constant tvhatever be the direction of QQ' . II. C. Ixvi THE EARLIER HISTORY OF CONICS. If BAB' be the particular segment whose vertex is A, so that BB' is bisected perpendicularly by the axis at the point If where A.y^PV, and if (JD be drawn perpendicular to PV, we have (by No. 3 on p. liii) Also, since AN = PV, QV : BN-=p :pa\ .•. BN=QD. Hence BN.AN=QD.PV, and AABB' = APQQ'. Therefore the triangle PQQ' is of constant area provided that FV is of given length. Also the area of the segment PQQ' is equal to ^. /\PQQ' ; [No. 7, p. Ixiii]. therefore the area of the segment is also constant under the same conditions. 10. The area of any ellipse is to that of a circle whose diameter is equal to the niajm' axis of the ellipse as the minor axis is to the rmtjor (or the diameter of the circle). [This is proved in Prop. 4 of the book On Conoids and Spheroids.] 11. The area of an ellipse wJwse axes are a, h is to that of a circle whose diameter is d, as ah to d^. [On Conoids and Spheroids, Prop. 5.] 12. The areas of ellipses are to one another as the rectangles under their axes ; and hence similar ellipses are to one another as the squares of corresponding axes. [On Conoids ami Spheroids, Prop. 6 and Cor.] It is not within the scope of the present Avork to give an account of the applications of conic sections, by Archimedes and others, e.g. for the purpose of solving equations of a degree higher than the second or in the problems known as vcuacts*. The former application * The word vtvci^, commonly inclinatio in Latin, is difficult to translate satisfactorily. Its meaning is best gathered from Pappus' explanation. He says (p. C70) : " A line is said to verge [vtvuv) towards a point if, being produced, it reaches the point." As particular cases of the general form of the problem he gives the following : ' ' Two lines being given in position, to place between them a straight line given in length and verging towards a given point." "A semicircle and a straight Hne at right angles to the base being given in ARCIIIMEDKS. IxvU is involved in Prop. 4 of Book IT. (hi thr Sp/it're aiifl Ci/Rii'ler, whore the problem is to cut a given sphere (by a plane) so that the segments may bear to one another a given ratio. The book On Spirals contains propositions which assume the solution of certain i'£vVct9, e.g. Props. 8 and 9, in which Archimedes a.ssumes the following problem to be eftected : If Λ Β be any chord of a circle and any point on the circumference, to draw through a straight line OBP meeting ΛΒ in D and the circle again in Ρ and such that DP is equal to a given length. Though Archimedes does not give the solution, we may infei• that he obtained it by means of conic sections*. A full account of these applications of conic sections by the (Greeks Λνϋΐ be found in the 11th and 12th chapters of Zeuthen's work. Die Lehre von den Kec/elschnitten im Alterhim. position, or two semicircles with their bases in a straight line, to place between the two lines a straight line given in length and verging towards a corner of the semicircle." Thus a line has to be laid across two given lines or curves so that it passes through a given point and the portion intercepted between the Unes or curves is equal to a given length. Zeuthen translates the word veOais by " Einschiebung, " or as we might say, "interpolation" ; but this fails to express the condition that the required line must pass through a given point, just as the Latin iuclhiatio (and for that matter the Greek term itself) does not explicitly express the other requirement that the intercepted portion of the line shall be of given length. * Cf. Pappus, pp. 298—302. PART Π. INTRODUCTION TO THE CONICS OF APOLLONIUS. CHAPTER I. THE AUTHOR AND HIS ΟλΥΝ ACCOUNT OF THE COXICS. We possess only the most meagre information about ApoUonius, viz. that he was born at Perga, in Pamphylia, in the reign of Ptolemy Euergetes (247-222 B.C.), that he flourished under Ptolemy Philopator, and that he went when quite young to Alexandria, where he studied under the successors of Euclid. We also hear of a visit to Pergamum, where he made the acquaintance of Eudemus, to whom he dedicated the first three of the eight Books of the Conies. According to the testimony of Geminus, quoted by Eutocius, he was greatly held in honour by his contemporaries, who, in admiration of his n)arvellous treatise on conies, called him the "great geometer*." Seven Books only out of the eight have survived, four in the original Greek, and three in an Arabic translation. They Λvere edited by Halley in 1710, the first four Books being given in Greek with a Latin translation, and the remaining three in a Latin translation from the Arabic, to which Halley added a conjectural restoration of the eighth Book. TJie first four Books have recently appeared in a new edition by J. L. Heiberg (Teubner, Leipzig, 1891 and 1893), wliich contains, in addition to the Greek text and a Latin translation, the fragments of the other works of ApoUonius wliich are still extant in Greek, the commentaries and lemmas of Pai)pus, and the commentaries of lOiitocius. • The quotation is from the sixth liook of Geminus' των μαθημάτων Οίωρία. See ApoUonius (ed. Heibein) Vol. ii. p. 170, THE AUTHOR AND HIS OWN ACCOUNT OF THE Coy/cs. Ixix Νυ iulditional light has been thrown on the Arabic text of Books V. to VII. since the monumental edition of Halley, except as regards the preface and the first few propositions of Book V., of which L. M. LudAvig Nix published a German translation in 1889*. For fuller details relating to the MSS. and editions of the Conies reference should be made to the Prolegomena to the second volume of Heiberg's edition. The following is a literal translation of the dedicatory letters in which Apollonius introduces the various Books of his Conies to Eudemus and Attalus respectively. 1. Book I. General preface. " Apollonius to Eudemus, greeting. " If you are in good health and circumstances are in other respects as you Avish, it is Avell ; I too am tolerably well. When I Avas with you in Pergamum, I observed that you Avere eager t(j become acquainted with my Avork in conies ; therefore I send you the first book which I have corrected, and the remaining books I Avill forward Avhen I have finished them to my satisfection. I daresay you have not forgotten my telling you that I undertook the investigation of this subject at the request of Naucrates the geometer at the time Avhen he came to Alexandria and stayed with me, and that, after Avorking it out in eight books, I communicated them to him at once, someAvhat too hurriedly, Avithout a thorough revision (as he was on the point of sailing), but putting doAvn all that occurred to me, Avith the intention of returning to them later. Wherefore I noAv take the opportunity of publishing each portion from time to time, as it is gradually corrected. But, since it has chanced that some other persons also Avho have been Avith me have got the first and second books before they Avere corrected, do not be surprised if you find them in a different shape. * This appeared in a dissertation entitled Das fiinfte Buck der Conica de» Apollonius von I'erga in der arabischcn Uebersetzung des Thabit ibn Corrah (Leipzig, 188'J), wbich however goes no further than the middle of the 7tb proposition of Book v. and ends ou p. 32 in the middle of a .sentence with thu words " gleich dem Quadrat von " ! The fragment is nevertheless valuable in that it gives a new translation of the important preface to Book v., part of which Halley appears to have misundorstood. Ixx INTRODUCTION TO APOLLONIUS. " Now of the eight books the first four form an elemeutary introduction ; the first contains the modes of producing the three sections and the opposite branches [of the hyperbola] (των avTLKei μίνων) and their fundamental properties worked out more fully and generally than in the writings of other authors ; the second treats of the properties of the diameters and axes of the sections as well as the asymptotes and other things of general imi)ortance and necessary for determining limits of pos- sibility (77/309 rov<i Βιορισμού^;)*, and what I mean by diameters and axes you will learn from this book. The third book contains many remarkable theorems useful for the synthesis of solid loci and determinations of limits; the most and * It is not possible to express in one word the meaning of διορισμοί here. In explanation of it it will perhaps be best to quote Eutocius who speaks of " that [διορισμοί] which does not admit that the proposition is general, but says when and how and in how many ways it is possible to make the required construction, like that which occurs in the twenty-second proposition of Euclid's Elements, From three stniinht lines, irJiich are equal to three {licen straight lines, to conntruct a triangle: for in this case it is of course a necessary condition that any two of the straight lines taken together must be greater than the remaining one," [Comm. on Apoll. p. 178]. In like manner Pappus [p. 30], in explaining the distinction between a 'theorem' and a 'problem,' says : " But he who propounds a problem, even though he requires what is for some reason impossible of realisation, may be pardoned and held free from blame ; for it is the business of the man who seeks a solution to determine at the same time [καΐ τοΐ'το δωρίσαι] the question of the possible and the impossible, and, if the solution be possible, when and how and in how many ways it is possible." Instances of the διορισμοί are common enough. Cf. Euclid vi. '27, which gives the criterion for the possibility of a real solution of the proposi- tion immediately following ; the διορισμοί there expresses the fact that, for a real solution of the equation .r((( - .v} = b-, it is a necessai-y condition that b -ψ- ( -\ . Again, we find in Archimedes, On the Sphere and Cylinder [p. 214], the remark that a certain problem " stated tiius absolutely requires a διορισμοί, but, if certain conditions here existing are added, it does not require a διορισμοί." Many instances will be found in Apollonius' work ; but it is to be observed that, as he uses the term, it frequently involves, not only a necessary condition, as in the cases just quoted, but, closely connected therewith, the determination of the number of solutions. This can be readily understood when the use of the word in tlie preface to Book iv. is considered. That Book deals with the number of possible points of intersection of two conies ; it follows that, when e.g. in the fifth Book hyperbolas are used for determining by their intersections with given conies the feet of normals to the latter, the number of solutions comes to light at the same time as the conditions necessary to admit of a solution. THE AUTHOR AND HIS OWN ACCOUNT OF THE t'OXICS. Ixxi prettiest of these theorems are new, and, when I had discovered thera, I observed that Euclid had not worked out the synthesis of the locus with respect to three and four lines, but only a chance portion of it and that not successfully: for it was not possible that the synthesis could have been completed without my additional discoveries. The fourth book shows in how many ways the sections of cones meet one another and the circumference of a circle : it contains other matters in ad<Jition, none of which has been discussed by earlier writers, concerning the number of points in which a section of a cone or the circumference of a circle meets [the opposite branches of a hyperbola] *. "The rest [of the books] are more by Avay of surplusage f (7Γ€ριουσιαστικωτ€ρα) : one of them deals somewhat fully (eVt TrXeov) with minima and maxima, one with equal and similar sections of cones, one with theorems involving determination of limits {Ζωριστίκών θ^ωρημ,άτων), and the last with determinate conic problems. * The reading here translated is Heiberg's κώνου τομή η κύκλου πΐριφέραα < rat's άντικειμέναΐί^- κατά, πόσα σημεία συμβάΧΚουσι. Halley had read κώνου τομή η κύκλου ττίριφέραα και (τι άντικ€ίμεναι άντικΐΐμέναΐί κατά πόσα σημάα συμβάλλουσι. Heiberg thinks Halley's longer interpolation unnecessary, but I cannot help thinking that Halley gives the truer reading, for the following reasons. (1) The contents of Book iv. show that the sense is not really complete without the mention of the number of intersections of a double-branch hyperbola with another double-branch hyperbola as well as with any of the single-branch couics ; and it is scarcely conceivable that AiJoUonius, in describing what was new in his work, should have mentioned only the less complicated question. (2) If Heiberg's reading is right we should hardly have the plural συμβάλλουσι after the disjunctive expres-;ion " a section of a cone or the circumference of a circle." (3) There is positive evidence for καΐ άντικΐΐ- μβΐΌΐ in Pappus' quotation from this preface [ed. Hultsch, p. 676], where the words are κώνου τομή κύκλου περιφερύψ και άντικΐίμΐναι άντικειμέναΐί, " a section of a cone with the circumference of a circle and opposite branches with opposite branches." Thus to combine the reading of our text and that of Pai)pus would give a satisfactory sense as follows : "in how many points a section of a cone or a circumference of a circle, as well as opposite branches, may [resiiectively] intersect opposite branches." See, in addition, the note on the corresponding passage in the preface to Book iv. given below. + πίριουσιαστικώτΐρα has baen translated " more advanced," but literally it implies extensions of the subject beyond the mere essentials. Hultsch translates "ad abundautiorem .scientiam pertinent," and Heiberg less precisely •'ulterius progrediuntur." Ixxii ixriiODUCTiox το apollonius. " When all the books arc published it will of course be open to those who read them to judge them cis they individually please. Farewell." 2. Preface to Book II. " Apoliouius to Eudenius, greeting. "If you are in good health, it is well; I too am moderately well. I have sent my son Apollonius to you with the second book of my collected conies. Peruse it carefully and com- municate it to those who are worthy to take part in such studies. And if Philonides the geometer, whom I introduced to you in Ephesus, should at any time visit the neighbourhood of Pergamum, communicate the book to him. Take care of your health. Farewell." 3. Preface to Book IV. " Apollonius to Attains, grec-ting. " Some time ago, I expounded and sent to Eudemus of Pergannim the first three books of my conies collected in eight books ; but, as he has passed away, I have resolved to send the remaining books to you because of your earnest desire to possess my Avorks. Accordingly I now send you the fourth book. It contains a discussion of the question, in how many points at most it is possible for the sections of cones to meet one another and the circumference of a circle, on the sup- position, that they do not coincide throughout, and further in how many points at most a section of a cone and the circum- ference of a circle meet the opposite branches [of a hyperbola] * • Here again Halley adds to the text as above translated the words και ^τι άντικύμίναι άντικαμ^ναΐί. Heiberg thinks the addition unnecessary as in the similar passage in the first ijreface above. I cannot but think that Halley is right both for the reasons given in the note on the earlier passage, and because, without the added words, it seems to me impossible to explain satis- factorily the distinction between the three separate questions referred to in the next sentence. Heiberg thinks that these refer to the intersections (1) of conic sections with one another or with a circle, (2) of sections of a cone with the double-branch hyperbola, (3) of circles with the double-branch hyperbola. But to specify separately, as essentially distinct questions, Heiberg "s (2) and ΤΙΙΚ AUTHOR AND HIS oWN ACCorXT OF THK (Ut.vjrs. Ixxiii and, besides these questions, not a few others of a similar character. Now the first-named ({iiestion Conon expounded to Thrasydaeus, without however showing proper mastery of the proofs, for which cause Nicoteles of Cyrene with some reason fell foul of him. The second matter has merely been mentioned by Nicoteles, in connexion with his attack upon Conon, as one capable of demonstration ; but I have not found it so de- monstrated either by himself or by any one else. The third (question and the others akin to it I have not found so much as noticed by any one. And all the matters alluded to, Avhich I have not found proved hitherto, needed many and various novel theorems, most of which I have already expounded in the first three books, while the rest are contained in the present one. The investigation of these theorems is of great service both for the synthesis of problems and the determinations of limits of possibility {ττρός re τάς των ττροβΧημάτων συνθύσας καΐ τον<; 8ίορισμού<;). On the other hand Nicoteles, on account of his controversy with Conon, Λνίΐΐ not have it that any use can be made of the discoveries of Conon for determinations of limits : in which opinion he is mistaken, for, even if it is possible, Avithout using them at all, to arrive at results re- lating to such determinations, yet they at all events afford a more ready means of observing some things, e.g. that several (3) is altogether inconsistent with the scientific method of Apollonius. When he mentions a circle, it is always as a mere appendage to the other carves {ύπίρβολη η i\\(i\j/ii rj κύκλου περιφέρεια is his nsual phrase), and it is impossible, I think, to imagine him drawing a serious distinction between (2) and (3) or treating the omission of Nicoteles to mention (3) as a matter worth noting, τό τρίτον should surely be something essentially distinct from, not a particular case of, TO δεύτερον. I think it certain, therefore, that το τρίτον is the case of the intersection of two double-branch hyperbolas with one another; and the adoption of Halley's reading would make the passage intelligible. We should then have the following three distinct cases, (1) the intersections of single-branch conies with one another or with a circle, (2) the intersections of a single-branch conic or a circle with the double- branch hyperbola, (3) the intersections of two double-branch hyperbolas ; and άλλο ούκ ολίγα δμοίο τούτοΐί may naturally be taken as referring to those cases e.f). where the curves toiicli at one or two points. Ixxiv INTRODUCTION TO Al'Ol.LONlUS. solutions are possible or that they are so many in number, and again that no solution is possible ; and such previous knowledge secures a satisfactory basis for investigations, while the theorems in question are further useful for the analyses of determinations of limits (ττρος τάς ανα\νσ€ΐς Be των 8io- ρισμων). Moreover, apart from such usefulness, they are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason." 4. Preface to Book V*. " Apollonius to Attalus, greeting. " In this fifth book I have laid down propositions relating to maximum and minimum straight lines. You must know that our predecessors and contemporaries have only superficially touched upon the investigation of the shortest lines, and have only proved what straight lines touch the sections and, con- vc'rsel}^ what properties they have in virtue of which they are tangents. For my part, I have proved these properties in the first book (without however making any use, in the proofs, of the doctrine of the shortest lines) inasmuch as I wished to place them in close connexion Avith that part of the subject in which I treated of the production of the three conic sections, in order to show at the same time that in each of the three sections numberless properties and necessary results appear, as they do with reference to the original (transverse) diameter. The propositions in which I discuss the shortest lines I have separated into classes, and dealt with each individual case by careful demonstration ; I have also connected the investigation of them with the investigation of the greatest lines above mentioned, because I considered that those who cultivate this science needed them for obtaining a knowledge of the analysis and determination of problems as well as for their synthesis, irrespective of the fact that the subject is one of those which seem worthy of study for their own sake. Farewell." * In the trauslution of this preface I have followed pretty closelj' the Geiiiiiiu translation of L. M. L. Nix above referred to [p. Ixix, note]. The prefaces to Books vi. and vii. are translated from Halley. THE AUTjfoR AND HIS OWN ACCOUNT oF THE ('OXICS. Ixxv 5. Preface to Book VI. " ApoUonius to Attains, greeting. " I send you the sixth book of the conies, which embraces propositions about conic sections and segments of conies et{ual and unequal, similar and dissimilar, besides some other matters left out by those who have preceded me. In particular, you will find in this book how, in a given right cone, a section is to be cut equal to a given section, and how a right cone is to be described similar to a given cone and so as to contain a given conic section. And these matters in truth I have treated somewhat more fully and clearly than those who wrote before our time on these subjects. Farewell." 6. Preface to Book VII. " ApoUonius to Attalus, greeting. " I send to you with this letter the seventh book on conic sections. In it are contained very many new propositions concerning diameters of sections and the figures described upon them ; and all these have their use in many kinds of problems, and especially in the determination of the conditions of their possibility. Several examples of these occur in the determinate conic problems solved and demonstrated by me in the eighth book, which is by way of an appendix, and which I will take care to send you as speedily as possible. Farewell." The first point to be noted in the above account by ApoUonius of his own work is tlie explicit distinction which he draws between the two main divisions of it. The first four Books contain matters wliich fall within the range of an elementary introduction (πίπτωκΐν CIS άγωγην στοιχειωδτ;), while the second four are extensions beyond the mere essentials (π^ριονσιαστικώτίρα.), οι• (as we may say) more "advanced,"' provided that we are careful not to undei-stand tlie relative terms "elementary" and "advanced" in the sense which we should attach to them in speaking of a modern mathematical work. Thus it would be wrong to regard the investigations of the fifth Book as more advanced than the earliei- Books on the ground that the results, leading to the determination of the evolute of any conic, are such as are now generally obtained by the aid of the IxXVi INTRODUCTION• TO APOLLOXIUS. differential calculus ; for the investigation of the limiting conditions for the possibility of drawing a certain number of normals to a given conic from a given point is essentially similar in character to many other διορισμοί found in other writers. The only difference is that, while in the case of the parabola the investigation is not very difficult, the corresponding propositions for the hyperbola and ellipse make exceptionally large demands on a geometer's acuteness and grasp. The real distinction between the first four Books and the fifth consists rather in the fact that the former contain a connected and scientific exposition of the general theory of conic sections as the indispensable basis for further extensions of the subject in certain special directions, Λvhile the fifth Book is an instance of such specialisation ; and the same is true of the sixth and seventh Books. Tlius the first four Books were limited to what were considered the essential principles; and their scope was that prescribed by tradi- tion for treatises intended to form an accepted groundwork for such special applications as were found e.g. in the kindred theory of solid loci developed by Aristaeus. It would follow that the subject- matter would be for the most part the .same as that of earlier treatises, though it would naturally be the object of Apollonius to introduce such improvements of method as the state of knowledge at the time suggested, with a view to securing greater generality and establishing a more thoroughly scientific, and therefore more definitive, system. One effect of the repeated working-up, by suc- cessive authors, of for the most part existing material Avould be to produce crystallisation, so to speak ; and therefore we should expect to find in the first four Books of Apollonius greater conciseness than would be possible in a treatise where new ground was being broken. In the latter case the advance would be more gradual, precautions would have to be taken with a view to securing the absolute impreg- nability of each successive position, and one result Avould naturally be a certain diffuseness and an apparently excessive attention to minute detail. We find this contrast in the two divisions of Apollonius' Conies; in fact, if we except the somewhat lengthy treatment of a small proportion of new matter (such as the properties of the hyperbola with two branches regarded as one conic), tiie first four Books are concisely put together in comparison with Books v.— VII. The distinction, therefore, between the two divisions of the work is the distinction between what may be called a text-book or com- \., THE AUTIIOli AND UTS OWN ACCOUNT OE THE OOXICK Ixxvii pendium of conic sections and a series of monographs on special portions of the subject. For the first four Books it Avill be seen tliat Apollonius does not chiim originality except as regards a number of theorems in the third Book and the investigations in the fourth Book about inter- secting conies ; for the rest he only claims that the treatment is more full and general than that contained in the earlier works on conies. This statement is quite consistent with that of Pappus that in his first four Books Apollonius incorporated and completed (αΐ'αττλτ^ρωσας) the four Books of Euclid on the same subject. Eutocius, however, at the beginning of his commentary claims more for Apollonius than he claims for himself. After quoting Geminus' account of the old method of producing the three conies from right cones Avith difierent vertical angles by means of plane sections in every case perpendicular to a genei'ator, he says (still purporting to quote Geminus), " But afterwards Apollonius of Perga investigated the general proposition that in every cone, whether right or scalene, all the sections are found, according as the plane [of section] meets tlie cone in difierent ways." Again he says, " Apollonius supposed the cone to be either right or scalene, and made the sections different by giving different inclinations to the plane." It can only be inferred that, according to Eutocius, Apollonius was the first discoverer of the fact that other sections than those perpendicular to a generator, and sections of cones other than right cones, had the same properties as the curves produced in the old way. But, as has already been pointed out, we find (1) that Euclid had already declared in the Phaenometia that, if a cone (presumably right) or a cylinder be cut by a plane not parallel to the base, the resulting section is a "section of an acute-angled cone," and Archimedes states expressly that all sections of a cone whicli meet all the generators (and here the cone may be oblique) are either circles or "sections of an acute-angled cone." And it cannot be supposed that Archimedes, or whoever discovered this proposition, could have discovered it otherwise than by a method which would equally show that hyperbolic and parabolic sections could be pro- duced in the same general manner as elliptic sections, which Archimedes singles out for mention because he makes special use of them. Nor (2) can any different conclusion be drawn from the continued use of the old names of the curves even after the more general method of producing them was known; there is nothing Ixxviii iNTRonrt iroN το apollonius. unnatural in this because, first, hesitation might well be felt in giving up a traditiijiiul definition associated with certain standard propositions, deterniinatif)ns of constiints, itc, and secondly, it is not thought strange, e.g. in a modern text-book of analytical geometry, to define conic sections by means of simple properties and equations, and to adhere to the definitions after it is proved that the curves represented by tlie general equation of the second degree are none other than the identical curves of the definitions. Hence we must conclude that the statement of Eutocius (which is in any case too general, in that it might lead to the supposition that every hyperlx)la could be produced as a section of any cone) rests on a misappre- hension, though perhaps a natural one considering that to him, living so much later, conies probably meant the treatise of Apollo- nius only, so that he might easily lose sight of the extent of the knowledge possessed by earlier writers*. At the same time it seems clear that, in the generality of his treatment of the subject from the very beginning, Apollonius was making an entirely new departure. Though Archimedes Λvas aware of the possibility of producing the three conies by means of sections of an oblique or scalene cone, we find no sign of his having used sections other than those which are perpendicular to the plane of synunetry ; in other words, he only derives directly from the cone the fundamental property referred to an axis, i.e. the relation ΓΝ' : AN. A'N^.P'N'°- : AN' . A'N', and Λνβ must assume that it was by means of the equation referred to the axes that the more general property QV : PV.P'V = (const) was proved. Apollonius on the other hand starts at once with * There seems also to have been some contusion in Eutocius' mind about the exact basis of tlic names panihohi, I'lUpse and hyperbola, though, as we .shall see, Apollonius makes this clear enough by connecting them immediately with the method of application of areax. Thus Eutocius speaks of the hyperbola as being so called because a certain pair of angles (the vertical angle of an obtuse-angled right cone and the right angle at which the section, made in the old way, is inclined to a generator) together exceed {ί'π(ρβά\\αν) two right angles, or because the iilane of the section passes beyond (ίηΓ(ρβά\\ίΐή the apex of the cone and meets the half of the double cone beyond the apex ; and he gives similar explanations of the other two names. But on this intei-pretation the nomenclature would have no significance ; for in each case we could choose different angles in the figure with equal reason, and so vary the names. THE AUTHOR AX]> HIS OWN ACfOUNT OF THF COXIf'S. Ixxix the most general section of an oblique cone, and proves directly from the cone that the conic has the latter general property with reference to a particular diameter arising out of his construction, which however is not in general one of the principal diameters. Then, in truly scientific fashion, he proceeds to show directly that the same property which was proved true with reference to the original diameter is equally true with reference to any other diameter, and the axes do not appear at all until they appear as par- ticular cases of the new (and arbitrary) diametei•. Another indica- tion of the originality of this fuller and more general Λvorking-out of the principal properties (τά αρχικά σνμπτωματα eVi irXiov και καθόλου μαΧλον ίξ(.φ-γασμΙνα) is, I tliiiik, to be found in the preface to Book V. as newly translated from the Arabic. ApoUonius seems there to imply that minimum straight lines (i.e. normals) had only been discussed by previous Avriters in connexion with the properties of tangents, whereas his own order of exposition necessitated an early introduc- tion of the tangent properties, independently of any questions about normals, for the purpose of eftecting the transition from the original diameter of reference to any other diameter. This is easily under- stood when it is remembered that the ordinary properties of normals are expressed with reference to the axes, and ApoUonius was not in a position to use the axes until they could be brought in as particular cases of the new and ai'bitrary diameter of reference. Hence he had to adopt a different order from that of earlier works and to postpone the investigation of normals for separate and later treatment. All authorities agree in attributing to ApoUonius the designation of the three conies by the names jjarabola, ellipse and hyperbola ; but it remains a question whether the exact form in which their fundamental properties were stated by him, and which suggested the new names, represented a new discovery or may have been known to earlier writers of whom Λνβ may take Archimedes as the repre- sentative. It will be seen from ApoUonius i. 11 [Prop. 1] that the fundamental property proved from the cone for the parabola is that expressed by the Cartesian equation y^-px, where the axes of coordinates are any diameter (as the axis of x) and the tangent at its extremity (as the axis of y). Let it be assumed in like manner for the ellipse and hyperbola that y is the ordinate drawn from any point to the original diameter of the conic, χ the abscissa mejvsured from one IXXX rXTRODUCTION TO APOLLOXIUS. extremity of the diameter, while .r, is tlie abscissa measured from the other extremity. Apollonius' procedure is then to take a certain length (/;, say) determined in a certain manner with reference to the cone, and to prove, frst, that y* : x.x,=p : (I (1), where d is the length of the original diameter, and, secondly, that, if a perpendicular be erected to the diameter at that extremity of it from which χ is measured and of length ]), then y- is equal to a rectangle of breadth χ and " applied " to the perpendicular of length p, but falling short (or exceeding) by a rectangle similar and similarly situated to that contained l)y ;j and d ; in other words, or 7/''=;λχ•+^.ατ' (2). Thus for the ellipse or hypei'lx)la an equation is obtained Avhich differs from that of the parabola in that it contains another term, and y* is less or greater than px instead of being equal to it. The line ρ is called, for all three curves alike, the parameter or latus rectum corresponding to the original diameter, and the characteristics expressed by the respective equations suggested the three names. Thus the parabola is the curve in which the rectangle which is equal to y^ is applied to ρ and neither falls short of it nor overlaps it, tlie ellipse and hyperbola are those in which the rectangle is applied t(j ]> but falls short of it, or overlaps it, respectively. In Archimedes, on the other hand, while the parameter duly appears with reference to the parabola, no such line is anywhere mentioned in connexion with the ellipse or hyperbola, but the fundamental property of the two latter curves is given in the form -JL• =-2^ X . a;, χ . a;,' ' it being fui-ther noted that, in the ellipse, either of the equal ratios b* . is equal to —^ in the case where the etjuation is referred to the axes and a, b ani the major and minor semi-axes respectively. Thus Apollonius' equation expressed the equality of two areas, while Archimedes' equation expressed the equality of two propor- THE AUTHOR AND HIS OWN ACCOUNT OF THE COXICS. Ixxxi tio7is ; and the question is whether Archimedes and his predecessors were acquainted with the equation of the central conic in the form in which ApoUonius gives it, in other words, whether tlie special use of the parameter or L•tus rectum for the purpose of graphically constructing a rectangle having χ for one side and equal in area to y- was new in ApoUonius or not. On this question Zeuthen makes the following observations. (1) The equation of the conic in the form had the advantage that the constant could be expressed in any shape which might be useful in a particular case, e.g. it might be expressed either as the ratio of one area to another or as the ratio of one straight line to another, in which latter case, if the consequent in the ratio were assumed to be the diameter d, the antecedent would be the parameter p. (2) Although Archimedes does not, as a rule, connect his description of conies Avith the technical expressions used in the well-knoAvn method of application of areas, yet the practical use of that method stood in the same close relation to the formula of Archimedes as it did to that of ApoUonius. Thus, where the axes of reference are the axes of the conic and a represents the major or transverse axis, the equation X. £C, (const.) = λ (say) is equivalent to the equation ^=^. = λ (3), ax + x ^ ' and, in one place {On Conoids and Spheroids, 25, p. 420) where Archimedes uses the property that — has the same value for all x.x^ points on a hyperbola, he actually expresses the denominator of the ratio in the form in Avhich it appears in (3), speaking of it as an area applied to a line equal to a but exceeding hy a square figure (ντΓΐρβάλλον ciSct τίτραγώ^ω), in other words, as the area denoted by ax + x^. (3) The equation — — = (const.) represents y as a mean pro- portional between χ and a certain constant multiple of x^, which H. C. / Ixxxii INTRODUCTION TO APOLLONIUS. last can easily be expressed as the ordinate Y, corresponding to the abscissa x, of a point on a certain straight line passing through the other extremity of the diameter (i e. the extremity from which a;, is measured). Whether this particular line appeared as an auxiliary line in the figures used by the predecessors of ApoUonius (of which there is no sign), or the well-known constructions were somewhat differently made, is immaterial. (4) The differences between the two modes of presenting the fundamental properties are so slight that we may regard Apollonius as in reality the typical representative of the Greek theory of conies and as giving indications in his proofs of the train of thought which had led liis predecessors no less than himself to the formulation of the various pjOpositions. Thus, where Archimedes chooses to use projwrtions in investiga- tions for Λvhich Apollonius prefers the method of application of areas which is more akin to our algebra, Zeuthen is most inclined to think that it is Archimedes who is showing individual peculi- arities rather than Apollonius, who kept closer to his Alexandrine predecessors : a view which (he thinks) is supported by the circumstance that the system of applying areas as found in Euclid Book II. is decidedly older than the Euclidean doctrine of pro- portions. I cannot but think that the argument just stated leaves out of account the important fact that, as will be seen, the Archimedean form of the equation actually appears as an intermediate step in the proof which Apollonius gives of his own fundamental equation. Therefore, as a matter of fact, the Archimedean form can hardly be regarded as a personal variant from the normal statement of the property according to the Alexandrine method. Further, to represent Archimedes' equation in the form ^ = (const.), X.Xi ^ ' and to speak of this as having the advantage that the constant may l)e expressed differently for different purposes, implies rather more than we actually find in Archimedes, who never uses the constant at all when the hyperbola is in question, and uses it for the ellipse only in the case where the axes of reference are the axes of the ellipse, and then only in the single form -= . α THE AUTH(1R AND HIS OWN ACCOUNT OF THE COXICS. Ixxxiii Now the equation _/_ = !' ax — x^ a- ' or y = ~ .X 4 . X , a a does not give an easy means of exhibiting the area y* as a simple rectangle applied to a straight line but falling short by another rectangle of equal breadth, unless we take some line equal to - and erect it perpendicularly to the abscissa χ at that extremity of it Avhich is on the curve. Therefore, for the purpose of arriving at an expression for y* corresponding to those obtained by means of the principle of application of areas, the essential thing was the determination of the parameter ρ and the expression of the con- stant in the particular form ^ , which however does not appear in Archimedes. Again, it is to be noted that, though Apollonius actually sup- plies the proof of the Archimedean form of the fundamental property in the course of the propositions i. 12, 13 [Props. 2, 3] establishing the basis of his definitions of the hyperbola and ellipse, he retraces his steps in i. 21 [Prop. 8], and proves it again as a deduction from those definitions : a procedure which suggests a somewhat forced adherence to the latter at the cost of some repetition. This slight awkwardness is easily accounted for if it is assumed that Apollonius was deliberately supplanting an old form of the fundamental property by a new one ; but the facts are more difiicult to explain on any other assumption. The idea that the form of the equation as given by Apollonius was new is not inconsistent with the fact that the principle of α]ψΙίοαίίοη of areas was older than the Euclidean theory of proportions ; indeed there would be no cause for surprise if so orthodox a geometer as Apollonius intentionally harked back and sought to connect his new system of conies with the most ancient traditional methods. It is curious that Pappus, in explaining the new definitions of Apollonius, says (p. 674) : " For a certain rectangle applied to a certain line in the section of an acute-angled cone becomes deficient by a square {Ιλλΐίττον τίτραγωνω), in the section of an obtuse-angled cone exceeding by a square, and in that of a right-angled cone neither deficient nor exceeding." There is evidently some confusion /2 IXXXIV INTRODUCTION TO APOLLONIUS. here, because in the definitions of Apollonius there is no question of exceeding or falling-short hy a square, but the rectangle which is equal to y* exceeds or falls short by a rectangle similar and similarly situated to that contained by the diameter and the latus rectum. The description "deficient, or exceeding, by a square" recalls Archimedes' description of the rectangle χ . .r, appearing in the equation of the liyperbola as νπ€ρβάΧλον ciSet τ€τραγωνω ; so that it would appear that Pappus somehow confused tlie two forms in which the two writers give the fundamental property. It will be observed that the " oppo.sites," by which are meant the opposite branches of a hyperbola, are specially mentioned as distinct from the three sections (the words used by Apollonius being των τριών τομών και των άντικίίμένων). They are first intro- duced in the proposition I. 14 [Prop. 4], but it is in i. 16 [Prop. 6] that they are for the first time regarded as together forming one curve. It is true that the preface to Book IV. shows that other writers had already noticed the two opposite branches of a hyper- bola, but there can be no doubt that the complete investigation of their properties was reserved for Apollonius. This view is supported by the following evidence. (1) The Avords of the first preface promise something new and more perfect with reference to the double-branch hyperbola as Avell as the three single-branch curves ; and a comparison between the works of Apollonius and Archimedes (who does not mention the two branches of a hyper- bola) would lead us to expect that the greater generality claimed by Apollonius for his treatment of the subject would show itself, if anywhere, in the discussion of the complete hyperbola. The words, too, about the "new and remarkable theorems" in the third Book point unmistakeably to the extension to the case of the complete hyperbola of such properties as that of the rectangles under the segments of intersecting chords. (2) That the treatment of the two branches as one curve was somewhat new in Apollonius is attested by the fact that, notwithstanding the completeness with which he establishes the correspondence between their properties and those of the single branch, he yet continues throughout to speak of them as two independent curves and to prove each proposition Λvith regard to them separately and subsequently to the demonstration of it for the single curves, the result being a certain diflTuseness which might have been avoided if the first propositions had been so combined as THE AUTHOR AND HIS OWN ACCOUNT OF THE <'(>\/cs. Ixxxv to prove each property at one and tlie same time for both double- branch and single-branch conies, and if the further developments had then taken as their basis the generalised property. As it is, the difluseness marking the separate treatment of the double hyperbola contrasts strongly with the remarkable ingenuity shown by ApoUonius in compressing into one proposition the proof of a property common to all three conies. This facility in treating the three curves together is to be explained by the fact that, as successive discoveries in conies were handed down by tradition, the general notion of a conic had been gradually evolved ; whereas, if ApoUonius had to add new matter with reference to the double hyperbola, it would naturally take the form of propositions supple- mentary to those affecting the three single-branch curves. It may be noted in this connexion that the proposition I. 38 [Prop. 15] makes use for the first time of the secondary diameter {d') of a hyperbola regarded as a line of definite length determined by the relation d^ _P d' " d' where d is the transverse diameter and ρ the parameter of the ordinates to it. The actual definition of the secondary diameter in this sense occurs earlier in tlie Book, namely between i. 16 and I. 17. The idea may be assumed to have been new, as also the determination of the conjugate hyperbola with two branches as the complete hyperbola which has a pair of conjugate diameters common with the original hyperbola, Λνΐίΐι the difference that the secondary diameter of the original hyperbola is the transverse diameter of the conjugate hyperbola and vice versa. The reference to Book II. in the preface does not call for any special remark except as regards the meaning given by ApoUonius to the terms diameter and axis. The Avords of the preface suggest that the terms were used in a new sense, and this supposition agrees with the observation made above (p. xlix) that Avith Archimedes only the axes are diameters. The preface speaks of the "many remarkable theorems" con- tained in Book III. as being useful for "the synthesis of solid loci," and goes on to refer more particularly to the "locus with respect to three and four lines." It is strange that in the Book itself we do not find any theorem stating in terms that a particular geometrical locus is a conic section, though of course we find Ixxxvi INTRODUCTION TO APOLLONIUS. theoi'ems stating conversely that all points on a conic have a certain property. The explanation of this is probably to be found in the fact that the determination of a locus, even when it was a conic section, was not regarded as belonging to a synthetic treatise on conies, and the ground for this may have been that the subject of such loci was extensive enough to require a separate book. This conjecture is supported by the analogy of the treatises of Euclid and Aristieus on conies and solid loci respectively, where, so far as we can judge, a very definite line of demarcation appears to have been drawn between the determination of the loci themselves and the theorems in conies Avhich were useful for that end. There can be no doubt that the brilliant investigations in Book V. with reference to normals regarded as maximuvi and minimum straight lines from certain points to the curve were mostly, if not altogether, new. It will be seen that they lead directly to the determination of the Cartesian equation to the evolute of any conic. Book VI. is about similar conies for the most part, and Book VII. contains an elaborate series of propositions about the magnitude of various functions of the lengths of conjugate diameters, including the determination of their maximum and minimum values. A comparison of the contents of Book VII. with the remarks about Book VII. and VIII. in the preface to the former suggests that the lost Book VIII. contained a number of problems having for their object the finding of conjugate diameters in a given conic such that certain functions of their lengths have given values. These problems would be solved by means of the results of Book VII., and it is probable that Halley's restoration of Book VIII. represents the nearest conjecture as to their contents which is possible in the present state of our knowledge. CHAPTER II. GENERAL CnARACTERISTICS. § 1. Adherence to Euclidean form, conceptions and language. The accepted form of geometrical proposition with whicli Euclid's Elements more than any other book has made mathematicians familiar, and the regular division of each proposition into its com- ponent parts or stages, cannot be better described than in the words of Proclus. He says*: "Every problem and every theorem which is complete with all its parts perfect purports to contain in itself all of the following elements : enunciation (ττρότασις), setting-out {Ικθίσι<;), definition^ (διορισμός), construction {κατασκ^νή), proof (άττόΒειξίς), conclusion {σνμ.τΓίρασμ.α). Now of these the enunciation states what is given and what is that which is sought, the perfect emmciation consisting of both these parts. The setting-out marks off" what is given, by itself, and adapts it beforehand for use in the investigation. The definition states separately and makes clear what the particular thing is which is sought. The construction adds what is wanting to the datum for the purpose of finding what is sought. The j^iOof draws the required inference by reasoning scientifically from ac- knowledged facts. The conclusion reverts again to the enunciation, confirming what has been demonstrated. These are all the parts of problems and theorems, but the most essential and those which are found in all are enunciation, proof, conclusion. For it is equally necessary to know beforehand Avhat is sought, and that tliis should be demonstrated by means of the intermediate steps and the de- monstrated fact should be inferred ; it is impossible to dispense • Proclus (ed. Friedlein), p. 203. t The word definition is used for want of a better. As will appear from what follows, διορισμό^ really means a closer description, by means of a concrete figure, of what the enunciation states in general terms as the property to be proved or the problem to be solved. Ixxxviii INTRODUCTION TO APOLLONIUS. with any of these three things. The remaining parts are often brought in, but are often left out as serving no purpose. Thus there is neither settitig-out nor definition in the problem of con- structing an isosceles triangle having each of the angles at the base double of the remaining angle, and in most theorems there is no construction because the setting-otit suffices without any addition for demonstrating the required property from the data. When then do Λνβ say that the setting-oui is wanting? The answer is, when there is nothing (jiven in the eyiunciation ; for, though the enun- ciation is in general divided into what is given and what is sought, this is not always the case, but sometimes it states only what is sought, i.e. what must be knoAvn or found, as in the case of the problem just mentioned. That problem does not, in fact, state beforehand with Λvhat datum Ave are to construct the isosceles triangle having each of the equal angles double of the remaining one, but (simply) that we are to find such a triangle.... When, then, the enunciation contains both (Avhat is given and what is sought), in that case Λνβ find both definition and setting-out, but, whenever the datum is wanting, they too are wanting. For not only is the setii7ig-out concerned with the datum but so is the definition also, as, in the absence of the datum, the definition will be identical with the enunciation. In fact, what could you say in defining the object of the aforesaid problem except that it is required to find an isosceles triangle of the kind referred to? But that is what the entmciation stated. If then the enunciation does not include, on the one hand, what is given and, on the other, what is sought, there is no setting-out in virtue of there being no datum, and the definition is left out in order to avoid a mere repetition of the enunciation." The constituent parts of an Euclidean proposition Λνϋΐ be readily identified by means of the above description without further details. It will be observed that the word διορισ /Aos has here a different .signification from that described in the note to p. Ixx above. Here it means a closer definition or description of the object aimed at, by means of the concrete lines or figures set out in the ίκθ(σί'; instead of the general terms used in the enunciation ; and its purpose is to rivet the attention better, as indicated by Proclus in a later passage, τρόπον TLva ττροσεχ^ίας ΙστΙν αΐτιοζ 6 διορισμός. The other technical use of the word to signify the limitations to which the possible solutions of a problem are subject is also described by Proclus, who speaks of διορισμοί determining " whether what is GENERAL CHAHACTEUISTICS. Ixxxix sought is impossible or possible, and ΙιΟΛν far it is practicable and in how many ways*"; and the διορισ/χος in this sense appears in the same form in Euclid as in Archimedes and Apollonius. In ApoUo- nius it is sometimes inserted in the body of a problem as in the instance ii. 50 [Prop. 50] given below ; in another case it forms the subject of a separate preliminary theorem, li. 52 [Prop. 51], the result being quoted in the succeeding proposition ii. 53 [Prop. 52] in the same way as the Stopta/xo's in Eucl. vi. 27 is quoted in the enunciation of vi, 28 (see p. cviii). Lastly, the orthodox division of a problem into analysis and synthesis appears regularly in Apollonius as in Archimedes. Proclus speaks of the preliminary analysis as a way of investigating the more recondite problems (τά άσαφίστερα των προβλημάτων) ; thus it happens that in this respect Apollonius is often even more formal than Euclid, who, in the Elements, is generally able to leave out all the preliminary analysis in consequence of the comparative sim- plicity of the problems solved, though the Data exhibit the method as clearly as possible. In order to illustrate the foregoing remarks, it is only necessaxy to reproduce a theorem and a problem in the exact form in which they appear in Apollonius, and accordingly the following propo- sitions are given in full as typical specimens, the translation on the right-hand side following the Greek exactly, except that the letters are changed in order to facilitate comparison Λvit^l the same propo- sitions as reproduced in this work and with the corresponding figures. III. 54 [Prop. 75 Avith the first figure]. Έά)/ κώνου τομής η κύκλου Trepi- If two straight hncs touching a φΐρΐίας δύο (νθ(Ίαι ΐφαπτόμΐναι συμ- section of a cune or the circum- πίπτωσι, δίά 8e των άφών παράλληλοι ference of a circle meet, and through άχθώσι Tois (φαπτομίναις, Koi άπί των the points of contact parallels be άφών npos TO avTo σημ(Ίοντης γραμμής drawn to the tangents, and from διαχθώσιν fxjOi'iai. τίμνουσαι τάς παραλ- the points of contact straight lines λήλους, TO ττΐρκχόμΐνον ορθογώνιον be drawn through the same point of ύπο των άποτ€μνομ(νων προς το άπο the curve cutting the parallels, the της ('πιζίυγνυοίσης τας άφας τΐτράγω- rectangle contained by the inter- vov λόγοι/ e';(et τον συγκείμΐνον tK Te ccpts bejirs to the square on the τοΰ, ov (χ(ΐ της ('πιζίυγνυούσης την line joining the points of contivct σϋμπτωσιν των (φαπτομίνων κα\ την the ratio compounded [1] of that 8ιχοτομίαντήςταςάφιις(πι.ζ(υγνυονσης which the square of tlie inner SOg- * Proclus, p. 202. xc INTRODUCTION TO APOLLONIUS. TO fVTos τμήμα προς το λοιπόν 8νναμ(ΐ, και τον, ορ €χ(ΐ το νπο των ίφαητομί- νων π(ρΐ(χόμ(νον όρθογωνιον npos το τίταρτον μίμος τον άπο ttJs Tas άφας (πιζ(ΐτγΐ'νοισης τ(τρα•γωνον. (στω κώνου τομή η κΰκ\ου π(ρι- φίρΐΐα ή ΑΒΓ κα\ (φαπτόμ(ναι αί ΑΔ, ΓΔ, και (πΐζ(νχθω ή ΑΓ κα\ 8ίχα Τίτμησθω κατά το Ε, κα\ (ΐν€ζ(νχθω η ΔΒΕ, κα\ ηχθω απο μΐν τοΐι Α πάρα την ΓΔ ί; ΑΖ, άπο δε τον Γ πάρα την ΑΔ ή ΓΗ, και (ΐΚήφθω τι σημύον eVt της -γραμμής το θ, κα\ ίπιζΐνχθΰσαι α'ι Αθ, Γθ (κβ(βλήσθωσαν (π\ τα Η, Ζ. λί'γω, ΟΤΙ το νπο ΑΖ, ΓΗ προς το άπο ΑΓ τον σνγκ(Ιμ{νον e;^e( λόγοι/ ίκ τοΐι, ον ίχ(ΐ το άπο ΕΒ προς το άπο ΒΔ κα\ το νπο ΑΔΓ προς το τίταρτον τον άπο ΑΓ, τοντϊστι το νπο ΑΕΓ. ηχθω yap άπο μ(ν τοΰ θ πάρα την AV ή ΚΘΟαΛ, από δί toG Β ί) ΜΒΝ • φαν(ρον 8ή, ΟΤΙ (φάπτ(ται η ΜΝ. ί'πίί ονν Ιση (στίν ή ΑΕ τή ΕΓ, ίση (στ\ κα\ ή MB τή Β Ν και ή KG τ^ ΟΛ κα\ ή ΘΟ τή OS και ή Κθ τή S\. ί'πίΐ ονν (φάπτονται α'ι MB, ΜΑ, κα\ πάρα την MB ηκται ή ΚΘΛ, ίστιν, ως τί) άπο AM προς το άπο MB, τοντϊστι ΤΙ) νπο ΜΒΝ, το άπο ΑΚ προς το νπο αΚΘ, τοντϊστι το νπο ΑΘΚ. ώς δί το νπο ΝΓ, ΜΑ προς το άπο ΜΑ, το νπο ΛΓ, ΚΑ προς το άπο ΚΑ• bi ίσον αρα, ως το ΰπί) ΝΓ, ΜΑ προς το νπο ΝΒΜ, το νπο ΑΓ, ΚΑ προς το νπο ment of the line joining the point of concourse of the tangents and the point of bisection of the line joining the points of contact bears to the square of the remaining seg- ment, and [2] of that which the rectangle contained by the tangents bears to the fourth part of the square on the line joining the points of contact. Let QPQ' be a section of a cone or the circumference of a circle and QT, Q'T tangents, and let QQ' be joined and bisected at V, and let TPV be joined, and let there be drawn, from Q, Qr parallel to Q'T and, from Q', Q'r' parallel to QT, and let any point R be taken on the curve, and let QR, (^R be joined and produced to /, r. I say that the rectangle contained by Qr, Q'r' has to the square on Q(/ the ratio compounded of that which the square on VP has to the square on PT and that which the rectangle under QTQ'*h!ifi to the fourth part of the square on QQ', i.e. the rect- angle under Q VQ'. For let there be dra\vn, from R, KRWR'K', and, from P, LPL' parallel to QQ' ; it is then clear that LL' is a tangent. Now, since QV is equal to VQ', LP is also equal to PL' and KW to WK' and R]V to WR' and KR to R'K'. Since therefore LP, LQ are tan- gents, and KRK' is drawn parallel to LP, as the square on QL is to the square on LP, that is, the rect- angle under LPL', so is the square on QK to the rectangle under R'KR, that is, the reotiingle under K'RK. And, as the rectangle under L'Q', * TO ύττό ΑΔΓ, "the rect. under QTQ'," means the rectangle QT. TQ', and similarly in other cases. GENERAL CHARACTERISTICS. ΛΘΚ. TO Se ΰπο ΑΓ, ΚΑ προς το νπο ΑΘΚ τον σνγκ(ίμ(νον (χ(ΐ λόγοι/ €Κ τοϋ της ΓΑ npos Αθ, τοντίστι της ΖΑ προς ΑΓ, και τον της ΑΚ προς Κθ, τοντίστί της ΗΓ πρ"ί ΓΑ, οί ίσην ό αντος τω, of c^fi το νπο ΗΓ, ΖΑ ττρόί το άπο ΓΑ• αίί αρα το νπο ΝΓ, ΜΑ ττροί το νπο ΝΒΜ, το ύπο ΗΓ, ΖΑ ττροΓ το άτΓο ΓΑ. το 8f νπο ΓΝ, ΜΑ προς το νπο ΝΒΜ τον νπο ΝΔΜ μίσον λαμβανομίνον τον σνγκ('ίμ(νον ^χα, λόγοι/ ί'κ τον, ον (χα το νπο ΓΝ, AM προς το νπο ΝΔΜ και το νπο ΝΔΜ προς το νπο ΝΒΜ • το αρα νπο ΗΓ, ΖΑ ττρο? το άπο ΓΑ τον σνγκίίμΐρον f\fi. Χογον (Κ τον τοΐ) νπο ΓΝ, AM προ? το νπο ΝΔΜ κα\ τοΐ) νπο ΝΔΜ προς το νπο ΝΒΜ. αλλ' ώς μίν το νπο Ν Γ, AM προΓ το νπο ΝΔΜ, το απο ΕΒ προς το άπο ΒΔ. ως be το νπο ΝΔΜ ττροί το νπο ΝΒΜ, το νπο ΓΔΑ προς το νπο ΓΕΑ• το αρα νπο ΗΓ, ΑΖ ττροΓ το άπο ΑΓ τον σνγκΐίμίνον ΐχιι λόγοι/ (Κ τον τον άπο BE προς το άπο ΒΔ και τον νπο ΓΔΑ προ: το νπο ΓΕΑ. LQ is to the square on LQ, so is the rectangle under K'<j', KQ to the square on KQ ; therefore c.v aerjuo i\s the rectiingle under L'(J\ LQ is to the rectangle under LTL, so i.s the rectangle under K'Q\ KQ to the rectangle under K'RK. But the rectangle under K'Q', KQ has to the rectiingle under K'RK the ratio compounded of that of Q'K' to A'7?, that is, oirQ to QQ', and of that of QK to A7i, that is, of r' ζ•' to Q'Q, which is the same as the ratio which the rectangle under r'Q', rQ has to the square on Q'Q; hence, as the rectangle mider L'Q', LQ is to the rectangle under L'PL, so is the rectangle under r'Q', rQ to the square on Q'Q. But the rectangle under Q'L', LQ has to the rectangle under L'PL (if the rectangle under L'TL be taken as a me;xn) the ratio compounded of that Avhich the rect- angle under Q'L', QL has to the rectangle under L'TL and the rect- angle imder L'TL to the rectangle imder L'PL; hence the rectangle under r'Q', rQ has to the square on Q'Q the ratio compounded of that of the rectangle under Q'L', QL to the rectangle under L'TI^ and of the rectangle under L'TL to the rectangle under L'PL. But, as the rectangle under L'Q', QL is to the rectangle under IJTL, so is the square on VP to the .square on PT, and, as the rectangle under L'TL is to the rectangle under Ζ7*Ζ, .so is the rectangle under Q'TQ to the rect- angle under (/ VQ; therefore the rect- angle under r'<^, rQ has to the .square on Q(^ the ratio compounded of that of the square on PV \x> the square on PT and of the rectangle under Q'TQ to the rectangle under Q' VQ. INTRODUCTION TO APOLLONIUS. II. 50 [Prop. 50 (Problem)]. (So far as relating to the hyperbola.) Ύηί 8<)θ(ίση! κωνον τομής (φαπτο- μ€νην άγαγήν, ήτις προς τω αξομι γωνίαν ιτοιήαα iVi ταντα τη τομή ισην ττ/ δο^ίΐ'σ/; οξ(ία γωνία. "Εστω ή τομή νπΐρβολή, και γ(γο- νίτω, κα\ «στω (φαπτομΐνη η ΓΔ, και (Ιληφθω το κίντρον τηί τομής το Χ, κα\ (π(ζ(νχβω ή ΓΧ κα\ κάθετος η ΓΕ• λόγοΓ αρα τον ΰττο των ΧΕΔ προς το απο της ΕΓ 8οθ(ίς• 6 αντος yap ΐστι τω της πλαγίας προς την ορβ'ιαν. τον be άπο της ΓΕ προς το άπο της ΕΔ λίίγοΓ fOTi 8οθΐίς• doOe'iaa γαρ ίκατίρα των νπο ΓΔΕ, ΔΕΓ. λόγος αρα κα\ τον νπο ΧΕΔ προς το άπο της ΕΔ δο^ίί'ί- ωστΐ κα\ της ΧΕ προς ΕΔ λόγοΓ (στϊ 8οθ€ίς. κα\ δοθί'ισα η προς τώ Ε • 8οθ(Ίσα αρα και η προς τω Χ. προς 8η θίσΐΐ (νθίία τη ΧΕ καΙ 8οθίντι τω Χ 8ιήκταί τις ή ΓΧ eV δ€8ομίνη γωνία- θίσΐΐ αρα η ΓΧ. θίσΐΐ δε και η τομή• δοθΐν αρα το Γ. και διήκται ίφαπτομίνη ή ΓΔ• θίσ^ι αρα η ΓΔ. ηχβω ασύμπτωτος της τομής ή Ζ\• ή ΓΔ (Ίρα (κβληθ(Ίσα σνμπΐσΰται τη άσνμπτωτω. σνμπιπτ(τω κατά το Ζ. μ(ίζων αρα (σται η νπο ΖΔΕ γωνία της νπί) ΖΧΔ. 8(ήσ(ΐ άρα (ΐς την σννθ(σιν την δΐ8ομ€νην οζ(Ίαν γωνιαν μ(ίζονα tivai τής ήμισίίης της π(ρΐ(χομίνης νπο των άσνμπτωτων. Το draw a tangent to a given section of a cone which shall make with the axis towards the same parts with the section an angle equal to a given acute angle. * # * ♦ Let the section be a hyperbola, and suppose it done, and let FT be the tangent, and let the centre C of the section be taken and let PC be joined and P^V be perpendicular ; therefore the ratio of the rectangle contained by CNT to the square on Λ^Ρ is given, for it is the same as that of the transverse to the erect. And the ratio of the square PN to the square on NT is given, for each of the angles PTJV, TNP is given. Therefore also the ratio of the rect- angle under CNT to the square on NT is given ; so that the ratio of CN to NT is also given. And the angle at Ν is given ; therefore also the angle at C is given. Thus with the straight line CN [given] in posi- tion and at the given point C a certain straight line PC has been drawn at a given angle ; therefore PC is [given] in position. Also the section is [given] in position ; there- fore Ρ is given. And the tangent Ρ Τ has been drawn ; therefore PT is [given] in position. Let the asymptote LC of the section bo drawn ; then PT pro- duced will meet the asymptote. Let it meet it in L ; then the angle LT^^ will be greater than the angle LCT. Therefore it will be necessary for the s^'uthcsis that the given acute angle should bo greater than GENERAL CHARAiTERISTICS. xcm σνντίθήσ(ται δη τί) προβ\ημα ού- τως- (στ<ύ η μ(ν bodflaa νπ€ρβολη, ής άξων ό ΑΒ, ασύμπτωτος 8ΐ ή ΧΖ, ή δί 8οθ(Ίσα γωνία οξύα μ(ίζων ούσα της ύτΓο των ΑΧΖ ή νττο ΚΘΗ, κ.α\ ίστω τί] νπο των ΑΧΖ Ίση ή νπο ΚΘΛ, κα\ ηχθω άπο τοΰ Α τϊ] ΑΒ προς ορθας ή ΑΖ, (Ιλήφθω δί τι σημά,ον trrt της Ηθ το Η, και ηχθω απ αύτον «πι την ΘΚ κάθΐτος η Η Κ. iVel ουν ΐση (στ\ν η νπο ΖΧΑ τη νπο ΛΘΚ, (Ισ\ 8( κα\ αΙ προς τοΊς Α, Κ γωνίαι ορθαΐ, ΐσην αρα, ως ή ΧΑ προς ΑΖ, ή ΘΚ προς ΚΑ. η δί ΘΚ προς ΚΑ μύζονα \oyov (χ(ί ηπ(ρ προς την ΗΚ • κα\ ή ΧΑ προ? ΑΖ αρα μείζονα \oyov (χ(ΐ ηπΐρ ή ΘΚ προς ΚΗ. ωστί κα\ το άπο ΧΑ πρόί το άπο ΑΖ μ(Ιζονα \oyov ΐχίί ηπ(ρ το άπο ΘΚ προς το άπο ΚΗ. αίί δί το άπο ΧΑ Trpof το άπο ΑΖ, τ; πλαγία προς την ορθΊαν κα\ ή πλαγία αρα προς την ορθίαν μείζονα \όγον e\fi ηπ^ρ το άπο ΘΚ προς το άπο ΚΗ. (αν 8η ποιήσωμΐν, ως το απο ΧΑ προς το άπο ΑΖ, όντως αΧλο τι προς το άπο ΚΗ, μΐ^ζον ΐσται τοϊι άπο ΘΚ. ίστω το νπο ΜΚΘ• κα\ ΐπίζ^νχθω η ΗΜ. ί'πίΐ ονν μΐ'ιζόν ΐστι τυ άπο Μ Κ του νπο ΜΚΘ, το αρα άπο Μ Κ ττροΓ το άπο ΚΗ μΐίζονα \όγον ίχ(ΐ ηπβρ το νπο ΜΚΘ προς το άπο ΚΗ, τοντίστι το άπο ΧΑ προς το άπο ΑΖ. και (άν ποιήσωμ€ν, ως το απο Μ Κ προς το απο ΚΗ, όντως το άπο ΧΑ προς αΧΧο τι, ίσται προς ίΧαττον τον άπο ΑΖ • και ή άπο τοΐι Χ ί'πΐ το \ηφθ(ν σημύον (πιζ(νγννμ(νη (νθΐΐα όμοια ποιήσΐΐ τα τρίγωνα, κα\ δια τοντο μ(ίζων (στιν ή νπο ΖΧΑ της νπο ΗΜΚ. κΐίσθω 8η ττ) νπο ΗΜΚ "ίση ή νπο ΑΧΓ• ή αρα ΧΓ τίμίΐ την τομήν. τίμνίτω κατά το Γ, κα\ άπο τοΐι Γ ϊφαπτομίνη τής τομής ηχθω ή ΓΔ, κα\ κάθίτος ή ΓΚ • ομοιον the half of that contained by the asymptotes. Thus the .synthesis of the prob- lem will proceed as follows : let the given hyperl>ola he that of which .LI' isthe axis and CZim asymptote, and the given acute angle (being greater than the angle ACZ) the angle FED, and let the angle FEII be equal to the angle ACZ, and let AZhe drawn from A at right angles to J.l', and let any point D be taken on DE, and let a perpendicu- lar I)F be drawn from it upon EF. Then, since the angle ZCA is equal to the angle ffEF, and also the angles a,t A, F are right, as CA is to AZ, so is EF to FIT. But EF has to FIT a greater ratio than it hiis to FD ; therefore also CA has to AZ a greater ratio than EF has to FD. Hence also the .square on CA has to the square on A Ζ a greater ratio than the square on EF has to the square on FD. And, as the square on C.i is to the square on AZ, so is the transverse to the erect ; therefore also the transverse has to the erect a greater ratio than the square on EF has to the square on FD. If then we make, as the square on CA to the square on AZ, so some other area to the square on FD, that area will be greater than the square on EF. Let it be the rectangle under KFE; and let Z) A' be joined. Then, since the square on KF is greater than the rectangle under KFE, the square on KF luis to the square on FD a greater ratio than the rectangle under KFE has to the square on FD, that is, the square on CA to the square on AZ. And if we make, as the .square on KF to the .siiuare on FD, so the .square on CA to INTRODUCTION TO APOLLONIUS. apa fWi TO ΓΧΕ τρίγωνου τω HMK. (στιν apa, ώς το άπυ ΧΕ πμοί το άπο ΕΓ, Γο άπο Μ Κ irpos το άπο ΚΗ. eoTt δί και, ως ή π\α•γία προς την ορθίαν, τό τ( νπο ΧΕΔ προς το απο ΕΓ και το νπο ΜΚΘ προς το άπο ΚΗ. Kcu άνάπα\ιν, ως το άπο ΓΕ προς το νπο ΧΕΔ, τό άπο ΗΚ προς το νπο ΜΚΘ• δι* ίσον άρα, ώς το άπο ΧΕ προς το νπο ΧΕΔ, το άπο Μ Κ προς το νπο ΜΚΘ. κα\ ώς αρα ή ΧΕ προς ΕΔ, ή ΜΚ προς Κθ. ην 8e κηι, ώς η ΓΕ προς ΕΧ, jJ ΗΚ προς KM • δι' ίσου αρα, α)Γ »; ΓΕ προς ΕΔ, 7 ΗΚ προ? Κθ. κα\ ΐΙσ\ν ορθα\ α'ι προς τοΙς Ε, Κ γωνίαι ■ Ίση αρα ή προς τω Δ γωνία τη νπο ΗΘΚ. another are;i, [the ratio] will be to a .smaller area than the square on AZ; and the straight line joining C to the point taken will make the triangles similar, and for this rciX-son the angle ZCA is greater than the angle DKF. Let the angle ACT be made equal to the angle DKF; therefore CP will cut the section. Let it cut it at P, and from Ρ let Ρ Τ be drawn touching the section, and 7*iV perpendicular ; therefore the triangle PCN is similar to DKF. Therefore, a.s is the square on CN to the square on NP, so is the square on KF to the square on FD. Also, as the transverse is to the erect, so is both the rectangle under CNT to the square on NP and the rectangle under KFE to the square on FD. And conversely, as the square on PN is to the rectangle under CNT, so is the square on DF to the rectangle under KFE; thereft)re ex aequo, as the square on CN is to the rectangle under CXT, so is the square on KF to the rectangle under KFE. There- fore, as CN is to NT, so is KF to FE. But also, as PN is to NC, so was DF to FK ; therefore ex aequo, as Ρ Ν is to NT, so is DF to FE. And the angles at N', F are right ; therefore the angle at Τ is equal to the angle DEF. In connexion with the propositions just quoted, it may not be out of place to remark upon some peculiar advantages of the Greek language as a vehicle for geometrical investigations. Its richness in grammatical forms is, from this point of view, of extreme import- ance. For instance, nothing could be more elegant than the regular u.se of the perfect imperative passive in constructions; thus, Avhere we should have to say " let a perpendicular be drawn " or, more peremptorily, "draw a perpendicular," the Greek expression is ηχθω GENERAL CIIAUACTERISTICS. XCV κάθΐτος, the former Avord expressing in itself the meaning " let it //are been drawn" or "suppose it drawn," and similarly in all other cases, e.g. •γ€•γράφθω, €π€ζευχθω, ίκβίβλησθω, Τίτμησθω, ίΐλτ;φ^ω, άφιψΊΐσθω and the like. Neatest of all is the word γεγονί'τω with which the analysis of a problem begins, " suppose it done." The same form is used very effectively along with the usual expression for a propor- tion, e.g. πίΤΓΟίησθω, ώς τ; HK ττρό? KE, η ΝΞ ττρο? EM, which can hardly be translated in English by anything shorter than " Let ΝΞ be so taken that ΝΞ is to ΞΜ as Η Κ to KE." Again, the existence of the separate masculine, feminine and neuter forms of the definite article makes it possible to abbreviate the expressions for straight lines, angles, rectangles and squares by leaving the particular substantive to be understood. Tims τ; Η Κ is 77 Η Κ (γραμμή), tJie line ΗΚ; ιχχ-η νπο ΑΒΓ or η νπο των ΑΒΓ the word understood is γωνία and the meaning is the aiujle ΑΒΓ (i.e. the angle contained by AB and ΒΓ) ; το νπο ΑΒΓ or το υπό των ΑΒΓ is το νττο ΑΒΓ (χωρίον or ορθο-γώνίον), the rectangle contained by AB, ΒΓ ; τό άπο AB is το αττό AB (τ^τράγωνον), tJie square on AB. The result is that much of the language of Greek geometry is scarcely less concise than the most modern notation. The closeness with which Apollonius followed the Euclidean tradition is further illustrated by the exact similarity of language between the enunciations of Apollonius' propositions about the conic and the corresponding propositions in Euclid's third Book about circles. The following are some obvious examples. Eucl. III. 1. Ap. II. 45, ToO δοθίντος κύκλου το κίντρον Της δοθίίσης Αλειψβωί η iVf/j- (ΰρύν. βοΧης το KfVTpov tvpuv. Eucl. in. 2. Αρ. I. 10. Έάν κύκλου fVi της 7repi0epiiaf Έαν ί'πι κωνον τομής Χηφθη δυο "Κηφθη δύο τυχόντα στ/μίΐα, ή fVi τα σημ(Ία, ή pev fVi τα σ•ημ(Ία (πιζίνγνυ- σημ(Ί.α (πιζίυγννμίνη evuela (ντος μίνη (vdeui (ντος πΐσΰται της τομής, πΐσίϊται τοΰ κύκλου. ή δι (π (ύθίίας αύτη (κτός. Eucl. ΠΙ. 4, Αρ. II. 26. Έάν «ν κύκλω δύο (ύθΐΐαι τίμνωσιν 'Εάν iv (Kkti^ft tj κύκλου ntpi- άλληλας μη δια τοΰ κίντρον ουσαι, ου φ(ρΐία δύο (ύθΰαί τίμνωσιν άλλήλας τίμνουσιν άλλήλας δίχα. μη δια τοΰ κίντρου ονσαι, ού τίμνονσιν άλλήλας δίχα. XCVl INTRODUCTION TO APOLLONIUS. Eucl. III. 7. Έαν κύκλου e»ri της διαμίτρον Χηφθή τι (τημΰον, ο μη (στι κίντρον του κΰκ\ου, άπο δε του σημύου προς τον κύκλοι/ προσπίιττωσιν (ΰθί'ιαί Tivts, μ('γίστη μ(ν (σται, ί'φ' ης το κίντρον, (ΧαχΙστη Se ή Χοιττη, των 8ΐ αΧΧων ae\ ή (yyiov της δια του κϊντρου της άπώτίρον μΐίζων (στίν, 8ύο 8e μόνον ΐσαι άπο τοΰ σημύου προσπ€σοΰνται προς τον κνκΧον (φ' ίκάτερα της (λαχίστης. Αρ. V. 4 and 6. (Translated from Halley.) If a point be taken on the axis of an ellipse whose distance from the vertex of the section is equal to half the latus rectum, and if from the point any straight lines what- ever be drawn to the section, the least of all the straight lines drawn from the given point will be that which is equal to half the latus rectum, the greatest the remaining part of the axis, and of the rest those which are nearer to the least will be less than those more re- mote As an instance of Apollonius' adherence to the conceptions of Euclid's Elements, those propositions of the first Book of the Conies may be mentioned which first introduce the notion of a tangent. Thus in I. 17 we have the proposition that, if in a conic a straight line be drawn through the extremity of the diameter parallel to the ordinates to that diameter, the said straight line will fall without the conic ; and the conclusion is drawn that it is a tangent. This argument recalls the Euclidean definition of a tangent to a circle as " any straight line which meets the circle and being produced does not cut the circle." We have also in Apollonius as well as in Euclid the proof that no straight line can fall between the tangent and the curve. Compare the following enunciations : Eucl. HI. 16. Ή τη 8ιαμ(τρω τοΰ κΰκΧου προς όρβας απ" άκρας ατγομίνη (κτος π^σύται τοΰ κΰκ\ον, και (Ις τον μ(ταζν τύπον της τ( (νθ(ί(ΐς και της π(ριφ(ρ(ίας ίτϊρα (νθ(ΐα ου πηρί/χττίσίίται. Αρ. Ι. 32. Έαν κώνου τομής 8ιά της κορυφής (ύθΰα πάρα Τΐταγμΐνως κατηγμίνην αχθτ), ίφάπτίται της τομής, και els τον μ(ταξυ τόπον της τ( κώνου τομής κα\ της (ύθίίας ίτίρα tvuda ου παρ(μ- πίσίΐται. Another instance of the orthodoxy of Apollonius is found in the fact that, when enunciating propositions as holding good of a circle as well as a conic, he speaks of " a hyperbola or an ellipse or the circumference of a circle," not of a circle simply. In this he follows the practice of Euclid based upon his definition of a circle as "a GENERAL CHARACTERISTICS, XCVll plane figure bounded by one line." It is only very exceptionally that the word circle alone is used to denote the circumference of the circle, e.g. in Euclid iv. 16 and Apollonius i. 37. § 2. Planimetric character of the treatise. Apollonius, like all the Greek geometers whose works have come doΛvn to us, uses the stereon\etric origin of the three conies as sections of the cone only so far as is necessary in order to deduce a single fundamental plane property for each curve. This plane property is then made the basis of the further development of the theory, Λνΐΰΰΐι proceeds without further reference to the cone, except indeed when, by way of rounding-ofl' the subject, it is considered necessary to prove that a cone can be found Avhich will contain any given conic. As pointed out above (p. xxi), it is probable that the discovery of the conic sections was the outcome of the attempt of Menaechmus to solve the problem of the two mean proportionals by constructing the plane loci represented by the equations ar - ay, y^ - bx, xy = ah, and, in like manner, the Greek geometers in general seem to have con- nected the conic sections with the cone only because it was in their view necessary to give the curves a geometrical definition expressive of their relation to other known geometrical figures, as distinct from an abstract definition as the loci of points satisfying certain conditions. Hence finding a particular conic was understood as being synonymous with localising it in a cone, and we actually meet with this idea in Apollonius i. 52 — 58 [Props. 24, 25, 27], where the problem of " finding" a parabola, an ellipse, and a hyperbola satisfying certain conditions takes the form of finding a cone of Avhich the required curves are sections. Menaechmus and his contemporaries would perhaps hardly have ventured, without such a geometrical defini- tion, to regard the loci represented by the three equations as being really curves. When however they were found to be producible by cutting a cone in a particular manner, this fact Λν38 a sort of guarantee that they Avere genuine curves ; and there was no longer any hesitation in proceeding with the further investigation of their properties in a plane, without reference to their origin in the cone. There is no reason to suppose that the method adopted in the Solid Loci of Aristaeus was diflferent. We know from Pappus that Aristaeus called the conies ])y their original names ; whereas, if (as H.C. ^''^^^^"γΓ"• •- ■ . U {UKIVERSITT. V5 .___>.lll XCviii INTRODUCTION TO APOLLONIUS. the title might be thought to imply) he had used in his book the methods of solid geometry, he would hardly have failed to discover a more general method of producing the curves than that implied by their old names. We may also assume that the other predecessors of Apollonius used, equally with him, the planimetric method ; for (1) among the properties of conies which were well-known before his time there are many, e.g. the asymptote-properties of the hyperbola, Λvhich could not have been evolved in any natural way from the consideration of the cone, (2) there are practically no traces of the deduction of the plane properties of a conic from other stereometric investigations, even in the few instances where it would have been easy. Thus it would have been easy to regard an ellipse as a section of a right cylinder and then to prove the property of conjugate diameters, or to find the area of the ellipse, by projection from the circular sections ; but this method does not appear to have been used. § 3. Definite order and aim. Some Avriters liave regarded the Conies as wanting in system and containing merely a bundle of propositions thrown together in a hap-hazard way without any definite plan having taken shape in the author's mind. This idea may have been partly due to the words used at the beginning of the preface, where Apollonius speaks of having put down everything as it occurred to him ; but it is clear that the reference is to the imperfect copies of the Books Avhich had been communicated to various persons before they took their final form. Again, to a superficial observer the order adopted in the first Book might seem strange, and so tend to produce the same impression ; for the investigation begins with the properties of the conies derived from the cone itself, then it passes to the properties of conjugate diameters, tangents, etc., and returns at the end of the Book to the connexion of particular conies with the cone, which is immediately dropped again. But, if the Book is examined more closely, it is apparent that from the beginning to the end a definite object is aimed at, and only such propositions are given as are necessary for the attainment of that object. It is true that they contain plane properties which are constantly made use of after- wards ; but for the time being they are simply links in a chain of proof loading to the conclusion that the parabolas, ellipses and hyperbolas which Apollonius obtains by any possible section of any GENERAL CHARACTKRISTIC'S. xcix kind of circular cone are identical with those which are produced from sections of cones of revolution. The order of procedure (leaving out unnecessary details) is as ίο11θΛν8. First, we have the property of the conic which is the equivalent of the Cartesian equation referred to the particular diameter which emerges from the process of cutting the cone, and the tangent at its extremity, as axes of coordinates. Next, we are introduced to the conjugate diameter and the reciprocal relation be- tween it and the original diameter. Then follow properties of tangents (1) at the extremity of the original diameter and (2) at any other point of the curve which is not on the diameter. After these come a series of propositions leading up to the conclusion that any new diameter, the tangent at its extremity, and the chords parallel to the tangent (in other words, the ordinates to the new diameter) have to one another the same relation as that subsisting between the original diameter, the tangent at its extremity, and the ordinates to it, and hence that the equation of the conic when referred to the new diameter and the tangent at its extremity is of the same form as the equation referred to the original diameter and tangent*. Apollonius is now in a position to pass to the proof of the proposition that the curves represented by his original definitions can be represented by equations of the same form with reference to reciangulm• axes, and can be produced by mean.s of sections of right cones. He proceeds to propose tlie problem "to find" a parabola, ellipse, or hyperbola, when a diameter, the angle of inclination of its ordinates, and the corresponding parameter are given, or, in other words, when the curve is given by its equation referred to given axes. "Finding" the curve is, as stated above, regarded as synonymous with determining it as a section of a right circular cone. This Apollonius does in two steps : he first assumes that the ordinates are at right angles to the diameter and solves the problem for this particular case, going back to the method followed in his original derivation of the curA'es from the cone, and not using any of the results obtained in the intervening plane investigations ; then, secondly, he reduces the case where the ordinates are not perpen- * The definiteness of the design up to this point is attested by a formal recapitulation introduced by Apollonius himself at the end of i. 51 and concluding with the statemt-nt that " all the properties which have been shown to be true with regard to the sections by reference to the original diameters will equally result when the other diameters are taken." 9^ C INTRODUCTION TO APOLLONIUS. dicular to tlie diaiiieter to tlie former case, proving by his procedure that it is always possible to draw a diameter which is at right angles to the chords bisected by it. Thus what is proved here is not the mere converse of the first propositions of the Book. If that had been all that Λνναβ intended, the problems would more naturally have followed directly after those propositions. It is clear, hoAvever, that the solution of the problems as given is not possible without the help of the intermediate propositions, and that Apollonius does in fact succeed in proving, concurrently with the solution of the problems, that there cannot be obtained from oblique cones any other curves than can be derived from right cones, and that all conies have axes. The contents of the first Book, therefore, so far from being a fortuitous collection of propositions, constitute a complete section of the treatise arranged and elaborated Avith a definite intention throughout. In like manner it will be seen that the other Books follow, generally, an intelligible plan ; as, however, it is not the object of this introduction to give an abstract of the work, the remaining Books shall speak for themselves. CHAPTER III. THE METHODS OF APOLLONIUS. As a preliminary to the consideration in detail of the methods era[)loyed in the Conies, it may be stated generally tliat they follow steadily the accepted principles of geometrical investigation which found their definitive expression in the Elements of Euclid. Any one who has mastered the Elements can, if he remembers Avhat he gradually learns as he proceeds in his reading of the Conies, understand every argument of which Apollonius makes use. In order, however, to thoroughly appreciate the whole course of his thought, it is necessary to bear in mind that some of the methods employed by the Greek geometers were much more extensively used than they are in modern geometry, and were consequently handled by Apollonius and his contemporary readers witli much greater deftness and facility than would be possible, without special study, to a modern mathematician. Hence it frequently happens that Apollonius omits an intermediate step such as a practised mathema- tician would now omit in a piece of algebraical work which was not intended for the mere beginner. In several such instances Pappus and Eutocius think it necessary to supply the omission by a lemma. § 1. The principal machinery used by Apollonius as well as by tlie earlier geometers comes under the head of what has been not inappropriately called a geometrical Algebra; and it will be convenient to exhibit the part which this plays in the Conies under the following important subdivisions. (1) The theory of proportions. This theory in its most complete form, as expounded in the fifth and sixth Books of Euclid, lies at the very root of tiie systeiu of Cll INTKonUCTIOX TO APOl.LONIUS. ApoUoiiius ; and a very short consideration suffices to show how far it is capable of being used as a substitute for algebraical operations. Thus it is obvious that it supplies a ready method of effecting the operations of multiplication and division. Again, suppose, for example, that we have a series in geometrical progression consisting of the terms a^, cti, a» ... η,ι, so that We have th \aj a^ V a„ Thus the continued use of the method of proportions enables an expression to be given for the sum of the geometrical series (cf. the summation in Eucl. ix. 35). (2) The application of areas. AVhether the theory of proportions in the form in Avhich Euclid presents it is due to Eudoxus of Cnidus (408 — 355 B.C.) or not, there is no doubt that the method of application of areas, to which allusion has already been made, was used much earlier still. AVe have the authority of the pupils of Eudemus (quoted by Proclus on Euclid I. 44) for the statement that "these propositions are the discoveries of the Pythagorean muse, the application of areas, their exceeding, and their falling short" (17 tc παραβολή τών χοφίων καΐ η νπΐρβολη κα\ η eX\enj/i<;), Avhere we find the very terms afterwards applied by Apollonius to the three conic sections on the ground of the corresponding distinction between their respective fundamental properties as presented by him. The problem in Euclid i. 44 is " to apply to a given straight line a parallelogram which shall be equal to a given triangle and have one of its angles equal to a given rectilineal angle." The solution of this clearly gives the means of addimj together or subtracting any triangles, parallelograms, or other figures which can be decomposed into triangles. Next, the second Book of Euclid (with an extension Λvhich is found in vi. 27 — 29) su^jplies means for solving the problems of modern algebra so long as they do not involve expressions above the second degree, and provided, so far as the solution of quadratic equations is concerned, that negative and imaginary solutions are excluded ; the only further qualification to be borne in mind is that, since negative magnitudes are not used in Greek geometry, THK MKTlloDS OF ΛΓ< )!,!,( )XIUS. ClU it is often necessary to solve a problem in two parts, with dillerent figures, where one solution by algebra would cover both cases. It is readily seen that Book ii. of the Elements makes it possible to multiply two factors with any number of linear terms in each ; and the compression of the result into a single product follows by the aid of the a]rplication-i\\QorQn\. That theorem itself supplies a method of dividing the product of any two linear factors by a third. The remaining operations for Avhich the second Book affords the means are, however, the most important of all, namely, (a) the iinding of a square whose area is equal to that of a given rectangle [ii. 14], which ])roblem is the equivalent of extract- ing the square root, or of the solution of a pure quadratic equation, (I)) the geometrical solution of a mixed quadratic equation, wliich can be derived from ii. 5, 6. In the first case {a) we produce the side Λ Β of the rectangle to E, making BE equal to BC ; then Λνβ bisect Λ Ε in F, and, Λvith F as centre and radius FE, draw a circle meeting CB produced in G. Then FG'^FB'+BG\ Also FG' = FE'^AB.BE^FB ', whence, taking away the common FB", BG- = AB.BE. This corresponds to the equation X* = (ώ •(1). and BG or χ is found. In the second case (6) we have, if A Β is divided cijually at C and unequally at Z>, A /J. Dli + CD' - Cn-. [Eucl. II. ').J Now suppose All -a, UB-x. CIV INTRODUCTION TO AJ'OLLONIUS. Tlien ax — a* = rect. A Η = the gnomon CMF. Thus, if the area of the gnomon is given (= h^, say), and if a is given (-^ AB), the problem of solving the equation ax — x° -b' is, in the language of geometry, " To a given straight line (a) to apply a rectangle which shall be equal to a given square {b') and άφοιβηΐ by a square,^' i.e. to construct the rectangle AJf. A Κ /-^^ c X D Β / Η / L Ο This simply requires the construction of a gnomon, ecjual in area to b", of which each of the outer sides is given ( CB, or - J . Now we know the area — (i.e. the square 67''), and we know the area of part of it, the required gnomon CMF {- ¥) ; hence we have only to find the difference between the two, namely the area of the square LG, in order to find CD which is equal to its side. This can be done by applying the Pythagorean proposition, i. 47. Simson gives the following easy solution in his note on vi. 28-29. Measure CO perpendicular to AB and equal to 6, produce OC to Ν so that ON ^ CB ί or - j , and with as centre and radius ON describe a circle cutting CB in D. Then DB (or x) is found, and therefore the rectangle AH. For AD.DJU Cir-^CB' ^OD' =^OC'- + CD\ whence AD.DB = OC\ or a£c-a;* = 6' (2). THE METHODS OF APOLLONIUS. CV It is clear that it is a necessary condition of tlic possibility of a eal solution that Ir must not be greater than ( ?: ) , and that tlu• geometrical solution derived from Euclid does not differ from our practice of soh'ing a quadratic by completing the square on the side containing the terms in a;' and ic*. To show how closely Apollonius keeps to this method and to the old terminology connected therewith, we have only to compare his way of describing the foci of a hyperbola or an ellipse. .He says, " Let a rectangle equal to one fourth part of the 'tiguTe' [i.e. equal to CB-] be applied to the axis at either end, for the hyperbola or the opposite brandies exceeding, but for the ellipse deficient, by a square " ; and the case of the ellijjse corresponds exactly to the solution of the equation just given. * It will be observed that, while in this case there are two geometrically real solutions, Euclid gives only one. It must not however be understood from this that he was unaware that there are two solutions. The contrary may be inferred from the proposition vi. 27, in which he gives the διορισμός stating the necessary condition corresponding to b-^l-\ ; for, although the separate treat- ment, in the text translated by Simson, of the two cases where the base of the applied parallelogram is greater and less than half the given line appears to be the result of interpolations (see Heiberg's edition. Vol. n. p. 161), the dis- tinction is perfectly obvious, and we must therefore assume that, in the case given above in the text, Euclid was aware that x = AD satisfies the equation as well as x — BD. The reason why he omitted to specify the former solution is no doubt that the rectangle so found would simply be an equal rectangle but on BD as base instead of AD, and therefore there is no real object in distinguishing two solutions. This is easily understood when we regard tlie equation as a statement of the problem of finding two quantities whose sum («) and product (//-) are given, i.e. as equivalent to the simultaneous equations x + y = a, x)j = b\ These symmetrical equations have really only one solution, as the two apparent solutions are simply the result of interchanging the values of .r and ij. This form of the problem was known to Euclid, as appears from Prop. 86 of the Data (as translated by Simson) : " If two straight lines contain a parallelogram given in magnitude, in a given angle ; if both of them together be given, they shall each of them be given." From Euclid's point of view the equations next referred to in the text x^i^ax = b'^ have of course only one solution. cvi INTKODUCrioN To ATOI-LOXIUS. Again, from the proposition in Euclid ii. 6, Λνο ha\e, if A Β is bisected at C and produced to JJ, AD.Dn + CB'^CD\ ο A C/ Β Κ L "Τ' Η Ε G F Let us suppose that, in Euclid's figure, AB - a, BD = x. Then AD.DB = ax + x\ and, if this is equal to b" (a given area), the solution of the equation ax + χ- — 1/ is equivalent to finding a gnomon equal in area to 6* and having as one of the sides containing the inner right angle a straight line equal to the given length CB or - . Thus Λve know ί - j and />', and we have to find, by the Pythagorean proposition, a square equal to the sum of two given squares. To do this Simson draws BO at right angles to vl^ and equal to 0, joins CO, and describes with centre C and radius CO a circle meeting A Β produced in D. Thus BD, or x, is found. Now AD. DB + CB--^ CD- = C0' = CB' + B0\ whence A1).DB = B0\ or «a; + ,ΐ* — 6*. This solution corresponds exactly to Apollonius' determination of the foci of the hyperhola. THK MKIHODS ηΐ• AI'OLI.OML'S. CVil The equation x' — ax = 6" can be dealt witli in a similar manner. If AB^a, and if wo suppose the problem solved, so that AD - X, then ,t• - — ax = AM = the gntnnon CMF, and, to find the gnomon, we have its area (ό'), and the area Cli' 1• (0 by w hich the trnomon diflers from CJ)'. Thus we can find D (and therefore AD, or x) by the same construction as in the case innnediately preceding. Hence Euclid has no need to treat this case separately, l)ecause it is the same as the preceding except that here χ is equal to AD instead of BD, and one solution can be derived frou) the other. So far Euclid has not put his propositions in the form of an actual solution of the quadratic equations referred to, though he has in ii. 5, 6 supplied the means of solving them. In vi. 28, 29 however he has not only made the problem more general by substituting for the sqttare by Avhich the required rectangle is to exceed or fall short a paraUelograni similar and similarly situated to a given parallelogram, but he has put the propositions in the form of an actual solution of the general quadratic, and has prefixed to the first case (the deficiency by a parallelogram) the necessary condition of possibility [vi. 27] corresponding to the obvious διορισ/Λ09 referred to above in connection with the equation ax — χ- = h'. Of the problems in vi. 28, 29 Simson rightly says " These two problems, to the first of which the 27th prop, is necessary, are the most general and useful of all in the elements, and are most frequently made use of by the ancient geometers in the solution of other problems ; and therefore are very ignorantly left out by Tacquet and Dechales in their editions of the Elements, who pretend that they are scarce of any use.* " * It is strange that, notwithstanding this observation of Sinisun's, the three propositions vi. 27, 28, 29 are omitted from Todhunter's Euchd, which contains a note to this effect : " We have omitted in the sixtli Book I'ropositious 27, 28, 29 and the first solution which Euchd gives of Proposition 30, as they appear now to be never required and have been condemned as useless by various modern commentators ; see Austin, Walker, and Lardner." I would suggest that all three propositions should be at once restored to the text-books of Euclid with a note explaining their mathematical significance. CVlll INTRODUCTION TO AI'OLLONIUS. The enunciations of these propositions are as follows* : VI. 27. " Of all the parallelograms ajrplied to the same straight line and deficient hij jmrallelogravis similar and similarly situated to that which is described upon the half of the line, that tchich is applied to the half and is similar to its defect, is greatest. VI. 28. " To a given straight line to apply a parallelogram equal to a given rectilineal figure and deficient by a jmralMogram similar to a given jiarallelogram : But the given rectilineal figure must not he greater than tlie parallelogram applied to half of the given line and similar to tloe defect. VI. 29. " To a given straight line to apply a parallelog7-am equal to a given rectiliiieal figure and exceeding by a parallelogra7n similar to a given one." Corresponding propositions are found among the Data of Euclid. Thus Prop. 83 states that, ^' If a parallelogram equal to a given space be applied to a given straight line, deficient by a parallelogi-am given in species, the sides of the defect are given," and Prop. 8-4 states the same fact in the case of an excess. It is worth while to give shortly Euclid's proof of one of these propositions, and vi. 28 is accordingly selected. κ Ν * The translation follows the text of Heiberg's edition of Euclid (Teubner, 1883-8). THE METHODS OF APOLLONIUS. CIX Let AJi be the given stniiglit line, C the given area, D the parallelogram to which the (Iffcct of th(> roquired parallelogram is to be similar. Bisect AB at JE, and on ΣΒ describe a parallelogram OEBF similar and similarly situated to D [by vi. 18]. Then, by the διορισμός [vi. 27], AG must be either equal to C or greater than it. If the former, the problem is solved ; if the latter, it follows that the parallelogram EF is greater than C. Now construct a parallelogram LKNM equal to the excess of EF over C and similar and similarly situated to D [vi. 25]. Therefore LKNM is similar and similarly situated to EF, while, if GE, LK, and GF, LM, are homologous sides respectively, GE>LK, and GF>LM. Make GX (along GE) and GO (along GF) equal respectively to LK^ LM, and complete the parallelogram XGOP. Then GPB must be the diagonal of the parallelogram GB [vi. 26]. Complete the figure, and we have EF = C + KM, by construction, and XO = KM. Therefore the difference, the gnomon EliO, is equal to C. Hence the parallelogram TS, which is equal to the gnomon, is equal to C. Suppose now that AB -a, SP = x, and that δ : c is the ratio of the sides KN, LK of the parallelogram LKNM to one another ; we then have, if m is a certain constant, TB = m . ax, b , = m.- χ-, c b . C so that ax — χ = — . c in Proposition 28 in like manner solves tln^ ('(juation b ο C ax + - X' = c m ex INTRODUCTION TO APOLLONIUS. If we compare these equations witli those by which Apollonius expresses the fundamental property of a central conic, viz. it is seen that the only difference is that ρ takes the place of a and, instead of any parallelogram whose sides are in a certain ratio, that particular similar parallelogram is taken whose sides are />, d. Further, Apollonius draws ;; at right angles to d. Subject to these differences, the phraseology of the Conies is similar to that of Euclid : the square of the ordinate is said to be equal to a rectangle "applied to "a certain straight line (i.e. ^;»), "having as its width " (πλατο5 Ιχαν) the abscissa, and " falling short (or exceeding) by a figure similar and similarly situated to that contained by the diameter and the parameter." It Λνίΐΐ be seen from what has been said, and from the book itself, that Apollonius is nothing if not orthodox in his adherence to the traditional method of application of areas, and in his manipula- tion of equations between areas such as are exemplified in the second Book of Euclid. From the extensive use Avhich is made of these principles we may conclude that, where equations between^ areas are stated by Apollonius without proof, though they are not immediately obvious, the explanation is to be found in the fact that his readers as well as himself Avere so imbued with the methods of geometrical algebra that they were naturally expected to be able to work out any necessary intermediate step for themselves. And, with regard to the manner of establishing the results assumed by Apollonius, we may safely infer, with Zeuthen, that it was the practice to prove them directly by using the procedtire of the second Book of the Elements rather than by such combinations and transformations of the results obtained in that Book as we find in the lemmas of Pappus to the propositions of Apollonius. The kind of result most frequently assumed by Apollonius is some relation between the products of pairs of segments of a straight line divided by points on it into a number of parts, and Pappus' method of proving such a relation amounts practically to the pro- cedure of modern algebra, whereas it is niore likely that Apollonius and his contemporaries would, after the manner of yeonietrical algebia, draw a figure showii;g the various rectangles and squares, and thence, in many cases by simple inspection, conclude e.g. that one rectangle is equal to the sum of two others, and so on. \ THE METHODS OF AI'Ol.LOXirS. CXI An instance will make this clear. In Apollonius in. 2G [Prop. 60] it is assumed that, if E, Λ, B, C, D he points on a line in the order named, and if AB = CD, then EC.EB = AB. BD + ED. ΕΛ. This appears at once if we set oft' EB' perpendicular and equal to EB, and Ε A' along EB' equal to Ε A, and if we complete the parallelograms as in the figure*. Similarly Eutocius' lemma to ill. 29 [Prop. 61] is more likely to represent Apollonius' method of proof than is Pappus' 6th lemma to Book III. (ed. Hultsch, p. 949). (3) Graphic representation of areas by means of aux- iliary lines. The Greek geometers were fruitful in devices for the compression of the sum or difference of the ai-eas of any rectilineal figures into a single area ; and in fact the Elements of Euclid furnish the means of effecting such compression generally. The Conies of Apollonius contain some instances of similar procedure which deserve mention for their elegance. There is, first, the representation of the area of the square on the ordinate y in the form of a rectangle whose base is the abscissa x. AVhile the procedure for this purpose is, in * On the other hand Pappus' method is simply to draw a line with points on it, and to proceed semi-algebraically. Thus in tliis case [Lemma 4 to Book ni., p. 947] he proceeds as follows, first bisecting BC in Z. CE.EB + BZ-^ = EZ\ DE.EA+AZ-^=EZ^, AZ-^ = CA.AB + BZ-. CE . EB + I}Z-^ = DE . EA + CA .AD + BZ\ CE.EB = DE . Ε A + CA.A B, and while It follows that whence (and CA = BD). CXll INTRODUCTION TO APOLLONIUS. form, closely connected with the traditional application of areas, its special neatness is due to the use of a certain auxiliary line. The Cartesian equation of a central conic referred to any diameter of length d and the tangent at its extremity is (if (/' be the length of the conjugate diameter) , d" -d" , and the problem is to express the right hand side of the equation in the form of a single rectangle xY, in other words, to find a simple construction fur }' where ^ d" _d" Apollonius' device is to take a length ρ such that ρ _ (Γ- d~'d'' (so that ρ is the parameter of the ordinates to the diameter of length d). If PP' be the diameter taken as the axis of x, and Ρ the origin of coordinates, he draws PL perpendicular to PP' and of length p, and joins P'L. Then, if PV = x, and if VB drawn parallel to PL meets P'L in R we have (using the figures of Props. 2, 3), by similar triangles, p_VB _ Vli d~ P'V~d + x' so that VP ^]) + - X = Y, and the construction for Υ is therefore effected. Again, in v. 1-3 [Prop. 81], another auxiliary line is used for expressing y"^ in the form of an area standing on a; as base in the particular case whei-e y is an ordinate to the axis. AM is drawn perpendicular to ΛΑ' and of length equal to ^ (where p„ is the parameter corresponding to the axis A A'), and CM is joined. Tf the urdinate 7W meets CM in //, it is then proved that »/ 2 (quadrilateral MA Xll). THE METHODS OF APOLLONirs. Cxiii Apollonius then proceeds in v. 9, 10 [Prop. 86] to give, by means of a second auxiliary line, an extremely elegant construction for an area equal to the difference between the squai-e on a normal PG and the square on P'G, where P' is any other point on the curve than P'. The method is as foUoAvs. If PN is the ordinate of P, measure XG along the axis away from the nearer vertex so that NG :CN^p^'.AA'[^ CB' : CA']. In the figures of Prop. 86 let PN produced meet CM in //, as before. GH is now joined and produced if necessary, forming the second auxiliary line. It is then proved at once that NG - Nil, and therefore that NG'- = 2 Δ NGH, and similarly that NV = 2 Δ N'GH'. Hence, by the aid of the expression for y^ above, the areas PG' and P'G' are exhibited in the figures, and it is proved that P'G' -PG' = 2A HKH', so that Λνβ have in the figures a graphic representation of the difference between the areas of the two squares effected by means of the two fixed auxiliary lines CM, GH. (4) Special use of auxiliary points in Book VII. The seventh Book investigates the values of certain quadratic functions of the lengths of any two conjugate diameters PP', DD' in central conies of different excentricities, with particular reference to the maximum and minimum values of those functions. The whole procedure of Apollonius depends upon the reduction of the ratio CP' : CJ)'^ to a ratio between straight lines MH' and Mil, where //, //' are fixed points on the transverse axis of the hyperbola or on either axis of the ellipse, and Μ is a variable point on the same axis determined in a certain manner with reference to the position of the point P. The proposition that PP" : DD" = MH' : Mil appears in vii. 6, 7 [Prop. 127], and the remainder of the Book is a sufticient proof of the effectiveness of this formula as the geometrical substitute for algebraical operations. The bearing of the proposition may be exhibited as follows, with the help of the notation of analytical geometry. If the axes of H. c. h INTRODUCTION TO APOLLONIUS. coordinates are the principal axes of the conic, and if a, h are the lengths of the axes, we have, e.g., in the case of the hyperhoL•, cp.,ci>' ^<"^-^-) -{(!)" -(!)■} CP*-GD* (ΪΗΪΤ where .'>•, y are the coordinates of P. Eliminating y by means of the equation of the curve, we obtain CP'-CD' Apollonius' procedure is to take a certain fixed point // on the axis whose coordinates are (A, 0), and a variable point Μ whose coordinates are {x , 0), such that the numerator and denominator of the last expression are respectively equal to 2ax', 2ah ; whence the fraction is itself equal to j , and we have and h _a'-b' (i)> 2 From (1) we derive at once "'=4.τ» (2). whence AH : A'll =¥ : or ^p^iAA'. THE METHODS OF APOLLOMUS. CXV Thus, to find J7, we have only to divide ΑΛ' in the ratio p„ : AA'. This is what is done in vii. 2, 3 [Prop. 124]. £1' is similarly found by dividing A'A in the same ratio 2\i '. AA', and clearly AH = A'H', A'H=AH'. Again, from (2), we have f , a\ a' In other Λvords, A A' \A'M=CT: CN or A'M:AM=CN:TN (3). If now, as in the figures of Prop. 127, Λνο draw AQ parallel to the tangent at Ρ meeting the curve again in Q, AQ is bisected by CP; and, since AA' is bisected at C, it follows that A'Q is parallel to CP. Hence, if QM' be the ordinate of Q, the triangles A'QM', CPN are similar, as also are the triangles AQM', TPN ; .•. A'M':AM'=CN:TN. Thus, on comparison with (3), it appears that Μ coincides with M' ; or, in other words, the determination of Q by the construction described gives the position of Af. Since now //, //', Μ are found, and x', h Λvere so determined that CP' + CD' x' GP'-CD'~ A' it follows that CP"" : CD'' = x' + h:x'-h, or PP" : DO" = MH' : MH. The construction is similar for the ellipse except that in that case ^^' is divided externally at H, H' in the ratio described. § 2. The use of coordinates. We have here one of the most characteristic features of the Greek treatment of conic sections. The use of coordinates is not peculiar to Apollonius, but it will have been observed that the same point of view appears also in the earlier Avorks on the subject. Thus Menaechmus used the characteristic property of the paraljola which we now express by the equation y' —px referred to rectangular axes. He used also the property of the rectangular hyperbola which is expressed in our notation by tlie equation xy = c*, where the axes of coordinates are the asymptotes. Λ2 CXVl INTRODUCTION TO APOLLONIUS. Archimedes too used the same form of equation for the parabola, while his mode of representing the fundamental property of a central conic ~ — = (const.) can easily be put into the form of the Cartesian equation. So Apollonius, in deriving the three conies from any cone cut in the most general manner, seeks to find the relation between the coordinates of any point on the curve referred to the original diameter and the tangent at its extremity as axes (in general oblique), and proceeds to deduce from this relation, when found, the other properties of the curves. His method does not essentially differ from that of modern analytical geometry except that in Apollonius geometrical operations take the place of algebraical calculations. We have seen that the graphic representation of the area of y- in the form of a rectangle on χ as base, Avhere (;r, y) is any point on a central conic, was effected by means of an auxiliary fixed line P'Z whose equation referred to PP', PL as rectangular axes is That an equation of this form between the coordinates x, Υ repre- sents a straight line we must assume Apollonius to have been aware, because we find in Pappus' account of the contents of the first Book of his separate work on plane loci the following proposition : " If straight lines be drawn from a point meeting at given angles two straight lines given in position, and if the former lines are in a given ratio, or if the sum of one of them and of such a line as bears a given ratio to the second is given, then the point will lie on a given straight line"; in other words, the equation x-\-ay = h represents a straight line, where a, b are positive. The altitude of the rectangle whose base is χ and whose area is equal to y^ is thus determined by a procedure like that of analytical geometry except that Υ is found by a geometrical construction instead of being calculated algebraically from the equation of the auxiliary line THE METHODS OF APOLLOXIUS. CXvii If it should seem curious that the .auxilitary line is determined with reference to an independent (rectangular) pair of coordinate axes diflferent from the oblique axes to which the conic is itself referred, it has only to be borne in mind that, in order to show the area y' as a rectangle, it was necessary that the angle between χ and }' should be right. But, as soon as the line P'L was once drawn, the object Λvas gained, and the subsidiary axes of coordinates Λvere forthwith dropped, so that there was no danger of confusion in the further development of the theory. Another neat example of the use of an auxiliary line regarded from the point of view of coordinate geometry occurs in i. 32 [Prop. 11], where it is proved that, if a straight line be drawn from the end of a diameter parallel to its ordinates (in other Avords, a tangent), no straight line can fall between the parallel and the curve. Apollonius first supposes that such a line can be drawn from Ρ passing through K, a point outside the curve, and the ordinate KQV is drawn. Then, if y', y be the ordinates of A', Q respectively, and χ their common abscissa, referred to the diameter and tangent as axes, we have for the central conic (figures on pp. 23, 24) ?/''>?/* or xY, where Υ represents the ordinate of the point on the auxiliary line PL before referred to corresponding to the abscissa χ (with PP , PL as independent rectangular axes). Let y'^ be equal to xY\ so that Y' > 7, and let Y' be measured along Υ (so that, in the figures referred to, VR - Y, and YS = Y'). Then the locus of the extremity of Υ for different \'alues of χ is the straight line P'L, and the locus of the extremity of Y' for different points Κ on PK is the straight line Pti. It follows, since the lines P'L, PS intersect, that there is one point (their intersection R') where F= Y', and therefore that, for the corresponding points Q', Μ on the conic and the supposed line PK respectively, y = y , so that Q', Μ are coincident, and accordingly PK must meet the curve between Ρ and A". Hence Ρ Κ cannot lie between the tangent and the curve in the manner supposed. Here then we have two auxiliary lines used, viz. Y^P+'^x, d and Υ = mx, CXVIU INTRODUCTION TO APOLLONIUS. where m is some constant ; and the point of intersection of PK and the conic is determined b}' the point of intersection of the two auxiliary lines ; only here again the latter point is found by a geometrical construction and not by an algebraical calculation. In seeking in the various propositions of Apollonius for the equivalent of the Cartesian equation of a conic referred to other axes different from those originally taken, it is necessary to bear in mind what has already been illustrated by the original equation which forms the basis of the respecti\'^e definitions, viz. that, where the equivalents of Cartesian equations occur, they appear in the guise of simple equations between areas. The book contains several such equations between areas which can either be directly expressed as, or split up into parts Avhich are seen to be, constant multiples of x^, xy, y^, X, and y, where x, y are the coordinates of any point on the curve referi'ed to different coordinate axes ; and we have there- fore the equivalent of so many different Cartesian equations. Further, the essential difference between the Greek and the modern method is that the Greeks did not direct their efibrts to making the fixed lines of the figure as few as possible, but rather to expressing their equations between areas in as short and simple a form as possible. Accordingly they did not hesitate to use a number of auxiliary fixed lines, provided only that by that means the areas corresponding to the various terms in cc^, xy, . . . forming the Cartesian equation could be brought together and combined into a smaller number of terms. Instances have already been given in which such compression is efiected by means of one or ϊλυο auxiliary lines. In the case, then, where ίΛνο auxiliary fixed lines are used in addition to the original axes of coordinates, and it appears that the properties of the conic (in the form of equations between areas) can be equally well expressed relatively to the two auxiliary lines and to the two original axes of reference, we have clearly Avhat amounts to a transformation of coordinates. § 3. Transformation of coordinates. A simple case is found as early as i. 15 [Prop. 5], where, for the ellipse, the axes of reference are changed from the original diameter and the tangent at its extremity to the diameter conjugate to the first and the corresponding tangent. This transformation may with sufficient accuracy be said to be effected, first, by a simple transference of the origin of coordinates from the extremity of the original diameter THE METHODS OF Al'OLLONIUS. Cxix to the centre of the ellipse, and, secondly, by moving the origin a second time from the centre to i), the end of the conjugate diameter. We find in fact, as an intermediate step in the proof, the statement of the property that {d being the original diameter and d' its conjugate in the figure of Prop. 5) (0 the rectangle RT.TE where x, y are the coordinates of the point Q Λvith reference to the diameter and its conjugate as axes and the centre as origin ; and ultimately the equation is expressed in the old form, only with d' for diameter and ρ for the corresponding parameter, where p' _d d' ρ ' The equation of the hyperbola as well as of the ellipse referred to the centre as origin and the original diameter and its conjugate as axes is at once seen to be included as a particular case in I. 41 [Prop. 16], which proposition proves generally that, if two similar pai-allelograms be described on CP, CV respectively, and an equi- angular parallelogram be described on QV such that QV is to the other side of the parallelogram on it in the ratio compounded of the ratio of CP to the other side of the parallelogram on CP and of the ratio;? : d, then the parallelogram on QV is equal to the diiierence between the parallelograms on CP, CV. Suppose now that the parallelograms on CP, CV are squares, and therefore that the parallelogram on (^ Γ is a rectangle ; it follows that „ fdy d , = S.y (1). Apollonius is now in a position to undertake the transformation to a different pair of axes consisting of any diameter whatever and the tangent at its extremity. The method which he adopts is to use the new diameter as what has been termed an auxiliary fixed line. It will be best to keep to the case of the ellipse throughout, in order to avoid ambiguities of sign. Suppose that the new diameter CQ meets the tangent at Ρ in E, as in the figure of l. 47 [Prop. 21]; CXX INTRODUCTION TO APOLLONIUS. then, if from any point R on the curve tiie ordinate 7? IF is draAvn to PP\ it is parallel to the tangent PE, and, if it meets CQ in F, the triangles CPE, CWF are similar, and one angle in each is that between the old and the new diameters. Also, as the triangles CPE, C WF are the halves of two similar parallelograms on CP, CW, -we can use the relation proved in i. 41 [Prop. 16] for parallelograms, provided that we take a triangle on R W as base such that R WP is one angle, and the side WU lying along WP is determined by the relation RW CP ρ WU~ ΡΕ' d' Apollonius satisfies this condition by draAving i2i7 parallel to QT, the tangent at Q. The proof is as follows. From the property of the tangent, i. 37 [Prop. 14], QV' Ρ cr. VT d' Also, by similar triangles, QV RW and QV cv~ PE ' CP' Therefore RW WU PE ' CP ρ -d' RW CP ρ ,^, . , , .. V or wn~ 'PF ' 1 ^ required relation). Thus it is clear that the proposition I. 41 [Prop. 16] is true of the three triangles CPE, CFW, RUW; that is, aCPE-ACFW=ARUW (2). It is now necessary to prove, as is done in i. 47 [Prop. 21], that the chord RR' parallel to the tangent at Q is bisected by CQ*, in order to show that R^ί is the ordinate to CQ in the same way as * This is proved in i. 47 [Prop. 21] as follows : Δ CPE - A CFW= A RUW. Similarly Δ CPE - aCF'W= aR'UW. By subtraction, F'WWF=R'W'WR, whence, taking away the figure R'WWFM from each side, aR'F'M=aRFM, and it follows that RM=R'M. THE METHODS OF APOLLONIUS. CXXl 72 TF is to Cr. It then follows that the two triangles 7? Γ 11', CFW have tlie same relation to the original axes, and to the diameter QQ', as the triangles RFM, CUM have to the new axes, consisting of QQ' and the tangent at Q, and to the diameter PP', respectively. Also the triangle CPE has the same relation to the old axes that the triangle CQT has to the new. Therefore, in order to prove that a like relation to that in (2) above holds between three triangles similarly determined with reference to CQ, the tangent at Q and the diameter ΓΓ', it has to be shown that Δ CQT- Δ CUM^ AEMF. The first step is to prove the equality of the triangles CPE, CQT, as to which see note on i. 50 [Prop. 23] and in. 1 [Prop. 53]. We have then, from (2) above, acqt-acfw^apuw, or the quadrilateral QTWF=ARUW, therefore, subtracting the quadrilateral MUWF from each side, Δ CQT- A CUM= A RMF, the property which it was required to prove. Thus a relation between areas has been found in exactly the same form as that in (2), but with QQ' as the diameter of reference in place of PP. Hence, by reversing the process, we can determine the parameter q corresponding to the diameter QQ', and so obtain the equation of the conic with reference to the new axes in the same form as the equation (1) above (p. cxix) referred to PP' and its conjugate ; and, when this is done, Λνβ have only to move the origin from C to ^ in order to effect the complete transformation to the new axes of coordinates consisting of QQ' and the tangent at Q, and to obtain the equation Now the original parameter ρ is determined with reference to the length {d) of PF by the relation Ρ - ^^' - ^^ ^- = ^^ — d~ GV.VT~CP' PT~ PT' d ' OP so that ρ - -pj, • ^PE ; CXXIl INTRODUCTION TO APOLLONIUS. and the corresponding Λ -alue for q should accordingly be given by the equation which Apollonius proves to be the case in i. 50 [Prop. 23]. No mention of the parabola has been made in the above, because the proof of the corresponding transformation is essentially the same ; but it may be noted here that Archimedes was familiar with a method of effecting the same transformation for the parabola. This has been already alluded to (p. liii) as easily deducible from the proposition of Apollonius. There is another result, and that perhaps the most interesting of all, which can be derived from the foregoing equations between areas. We have seen that Δ7?ί/ΤΓ= Δ,ΟΡΕ- aCFJV, so that AEUW+ aCFW= aCPjE, i.e. the quadrilateral CFRU^ ACPE. Now, if PP', QQ' are fixed diameters, and R a variable point on the curve, we observe that RU, RF are drawn always in fixed directions (parallel to the tangents at Q, Ρ respectively), Avhile the area of the triangle CPE is constant. It follows therefore that, if PP, QQ' are two fixed diameters and if from any point R on the curve ordinates be dravm to PF, QQ' meeting QQ', PP in F, U respectively, then the area of the quadrilateral CFRU is constant. Conversely, if in a quadrilateral CFRU the ttvo sides CU, CF lie along fixed straight lines, ivhile the two other sides are drawn from a moveable jjoint R in given directions ami meeting the fixed lines, and if the quadrilateral has a constant area, then the locus of the j)oint R is an ellipse or a hyperbola. Apollonius does not specifically give this converse proposition, nor in fact any proposition stating that this or that locus is a conic. But, as he says in his preface that his work contains " remarkable theorems which are useful for the synthesis of solid loci," we must conclude that among them was the proposition which in effect states that the area of the quadrilateral CFRU is constant, and that the converse way of stating it was perfectly well known to him. THE METHODS OF APOLLONIUS. CXXlll It will be seen from the note to Prop. 18 that the proposition that the area of GFRU is constant is the equivalent of saying that the equation of a central conic referred to any two diameters as axes is ax' +βχ7/ + γΐ/ = Α, ■where a, β, y, A are constants. It is also interesting to observe that this equation is the equiva- lent of the intermediate step in the transformation from one diameter and tangent to another diameter and tangent as axes ; in other Avords, Apollonius passes from the equation referred to one pair of conjugate diameters to the equation referred to a second 2)<^i'>' of conjugate diameters hij means of the more general equation of the cu7've referred to axes consisting of one of each pair of conjugates. Other forms of the equation of the conic can be obtained, e.g. by regarding RF, JiU as fixed coordinate axes and expressing the constancy of the area of the quadrilateral CF'R'U' for any point R' with reference to RF^ RU as axes. The axes of reference may then be any axes meeting in a point on the curve. For obtaining the equation we may use the formula CFRU^ CF'R'U', or the other relations derived immediately from it, viz. F'lRF^ lUU'R', or FJR'F'^JU'UR, which are proved in iii. 3 [Prop. 55]. The coordinates of R' would in this case be R'l, R'J. Similarly an equation can be found corresponding to the property in III. [Prop. 54] that Δ HFQ = quadrilateral IITUR. Again, in. 54, 56 [Prop. 75] lead at once to the "locus \nth respect to three lines," and from this we obtain the well-known equation to a conic with reference to two tangents as axes, where the lengths of the tangents are h, k, viz. and, in the particular case of the parabola, ©'HD' CXXIV INTRODUCTION TO APOLLONIUS. The latter equation can also be derived directly from in. 41 [Prop. 65], which proves that three tangents to a parabola forming a triangle are divided in the same proportion. Thus, if X, y be the coordinates of Q with reference to qR, qP as axes, and if qp = x^, rq =?/, (cf. the figure of Prop. 65), we have, by the proposition, X rQ _yx-y _^-yx ^1 x^-x ~Qp y y. h-x/ From these equations we find X, X 2/. y -1 = ^-1, or x,^^ k,: ky Also, since -. y. ^ ^ y,~y ^■.^ = 1 ^, 2/, therefore by combining (1) and (2) we obtain ( il• (!)'-■ ;i)• (2). The same equation can equally be derived from the property proved by Archimedes (pp. lix, Ix). Lastly, we find of course the equation of the hyperbola referred to its asymptotes xy = c-, and, if Apolloaius had had a relation between the coordinates of a point (x, y) represented to him in a geometrical form equivalent to the equation xy Λ- ax + by Λ- C = Q, he Λvould certainly not have failed to see that the locus Avas a hyperljola ; for the nature of the equation would immediately have suggested the compression of it into a form which would show that the product of the distances of the point (reckoned in fixed directions) from twu fixed straight lines is constant. THE METHODS OF APOLT-OXIUS. CXXV § 4. Method of finding two mean proportionals. It will be remembered that Menaechinus' solution of the problem of the two mean proportionals was eifected by finding the points of intersection between any two of the curves .r* = ay, y^ =^bx, xy = ah. It is clear that the points of intersection of the first two curves lie on the circle x^ + y' — bx — ay = 0, and therefore that the two mean proportionals can be determined by means of the intersection of this circle with any one of the three curves. Now, in the construction for two mean proportionals which is attributed to ApoUonius, we find this very circle used, and we must therefore assume that he had discovered that the points of inter- section of the two parabolas lay on the circle. We have it on the authority of loannes Philoponus* (who quotes one Parmenio) that ApoUonius solved the problem thus. Let the two given unequal straight lines be placed at right angles, as 0Λ, OB. Complete the parallelogram and draw the diagonal OC On OC as diameter describe the semicircle OBC, produce OA, OB, and through C draw DCFE (meeting OA in D, the circle again in F, and OB in E) so that DC ^ FE. ''And this is assumed as a postulate unjn'oved." Now DC=FE, and therefore DF= CE. * On the Anal. post. Vol. II. p. 105. The passage is quoted iu Heiberg's Apolhnitts, CXXVl INTRODUCTION TO APOLLONIUS. And, since the circle on OC as diameter passes through A, OD.DA=FD.DC = CE.EF = OE.EB; .•. OD:OE = BE:AD (1). But, by similar triangles, OD:OE=CB:BE = OA:BE (2). Also, by similar triangles, OJ):OE = OA: AC = ΌΑ:ΟΒ (3). It follows from (1), (2) and (3) that OA:BE = BE:AD = AD.OB', hence BE, AD are the two required mean proportionals. The important step in the above is the assumed step of drawing DE through C so that DC = FE. If we compare with this the passage in Pappus Avhich says that ApoH'onius "has also contrived the resolution of it by means of the sections of the cone*," we may conclude that the point F in the above figure was determined by draAving a rectangular hyperbola with OA, OB as asymptotes and passing through C. And this is the actual procedure of the Arabian scholiast in expounding this solution. Hence it is sufficiently clear that Apollonius' solution Avas obtained by means of the intersection of the circle on OC as diameter with the rectangular hyperbola referred to, i.e. by the intersection of the curves o:^ + y^ — bx — ay ■ xy The mechanical solution attributed to Apollonius is given by Eutociust. In this solution M, the middle point of OC, is taken, and with 3i as centre a circle has to be described cutting OA, OB produced in points D, Ε such that the line DE passes through C ; and this, the writer says, can be done by moving a i-^der about C as a fixed point until the distances of D, Ε (the points in which it crosses OA , OB) from Μ are equal. * Pappus in. p. 56. Ούτοι ycip 6μo\oyo0ιn■es CTepebv elvai το πρόβλημα την κατασκΐνην αύτοΰ μόνον opyaviKuii πεποίηνται σνμφώνωί Άπό\\ωνΙψ τψ 11(ρ•γαΙψ, δί καΐ την άνάλυσιν αύτοΰ ττΐποίηται δια των τον κώνου τομών. t AicLiniedes, Vol. in. pp. 7G— 78. " 1 THE METHODS OF APOLLONIUS. CXXvii It is clear that this solution is essentially the same as the other, because, if DC be made equal to FE as in the former case, the line from J/ perpendicular to DE nmst bisect it, and therefore MD = ME. This coincidence is noticed in Eutocius' description of the solution of the problem by Philo Byzantinus. This latter solution is the same as that attributed by loannes Philoponus to Apollonius except that Philo obtains the required position for DE by mov-ing the ruler about C until DC, FE become equal. Eutocius adds that this solution is almost the same as Heron's (given just before and identical with the niechanical solution of Apollonius), but that Philo's method is more conΛ'enient in practice (ττρο? χρησιν (νθίτω- Tcpov), because it is, by dividing the ruler into equal and con- tinuous parts, possible to watch the equality of the lines DC, FE with much greater ease than to make trial with a pair of compasses (καρκίνω διαπ«ρά^€ΐν) whether MD, ME are equal*. It may be mentioned here that, when Apollonius uses the problem of the two mean proportionals in the Conies, it is for the purpose of connecting the coordinates of a point on a central conic with the coordinates of the corresponding centre of curvature, i.e. of the corresponding point on the evolute. The propositions on the subject are v. 51, 52 [Prop. 99]. § 5. Method of constructing normals passing through a given point. Without entering into details, for Λvhich reference should be made to v. 58-63 [Props. 102, 103], it may be stated generally that Apollonius' method of finding the feet of the various normals passing through a given point is by the construction of a certain rectangular hyperbola Λvhich determines, by its intersections with the conic, the required points. The analytical equivalent of Apollonius' procedure is as follows. Suppose to be the fixed point through which the normals are to pass, and FGO to be one of those normals, meeting the major or transverse axis of a central conic, or the axis of a parabola, in G. Let FN be the ordinate of F, and OM the perpendicular from on the axis. Then, if we take as axes of coordinates the axes of the central conic, and, for the parabola, * Archimedes, Vol. iii. p. 70. CXXVlll INTRODUCTION TO APOLLONIUS. the axis and the tangent at the vertex, and if (x, y), («,, y^ be the coordinates of P, respectiA'ely, we have y ^ NG — y, £c, — X - NG ' Therefore, (1) for the parabola, Pa y -"' -,—';- xy {^.-f)y-y.-^j = o (1); (2) for the ellipse or hyperbola, ό^ b' xy {l+^)-x,y±-..y,x = 0. The intersections of these rectangular hyperbolas with the respective conies give the feet of the various normals passing through 0. Now Pappus criticises this procedure, so far as applied to the ;;α?'α- bola, as being unorthodox. He is speaking (p. 270) of the distinction between the three classes of "plane" (tTriVcSa), "solid" (στ€ρ€ά), and the still more complicated " linear" problems (-γραμμικά προβλήματα), and says, " Such procedure seems a serious error on the part of geometers Avhen the solution of a plane problem is discovered by means of conies or higher curves, and generally when it is solved by means of a foreign kind (e^ ανοικείου yeVovs), as, for example, the problem in the fifth Book of the Conies of Apollonius in the case of the parabola, and the solid vcwts with reference to a circle assumed in the book about the spiral by Archimedes ; for it is possible without the use of anything solid to discover the theorem pro- pounded by the latter...." The first allusion must clearly be to the use of the intersections of a rectangular hyperbola with the parabola when the same points could be obtained by means of the intersec- tions of the latter with a certain circle. Presumably Pappus regarded the parabola itself as being completely drawn and given, so that its character as a " solid locus " was not considered to affect the order of the problem. On this assumption the criticism has no doubt some force, because it is a clear advantage to be able to effect the construction by means of the line and circle only. THE METHODS OF APOLLONIUS. CXXIX The circle in this case can of course be obtained l)y c<jmbining the equation of the rectangular hyperbola (1) above with that of the parabola y' — ρ,,Χ• Multiply (1) by — , and we have and, substituting p^^x for if, -■-(x,-'|).^-f =0, whence, by adding the equation of the parabola, we have But there is nothing in the operations leading to this result which could not have been expressed in the geometrical language which the Greeks used. Moreover we have seen that in Ajiollonius' solution of the problem of the two mean proportionals the same reduction of the intersections between two conies to the intersec- tions of a conic and a circle is found. We must therefore assume that Apollonius could have reduced the problem of the normals to a parabola in the same way, but that he purposely refrained from doing so. Two explanations of this are possible; either (1) he may have been unwilling to sacrifice to a pedantic orthodoxy the convenience of using one uniform method for all three conies alike, or (2) he may have regarded the presence of one "solid locus" (the given parabola) in his figure as determinative of the class of problem, and may haA'e considered that to solve it with the help of a circle only would not, in the circumstances, have the effect of making it a " plane " problem. H. C. CHAPTER IV. THE CONSTRUCTION OF A CONIC BY MEANS OF TANGKNTS. In Book III. 41-43 [Props. G5, 66, 67] Apollonius gives three theorems which may be enunciated as follows : 41. If three straight lines, each of which totiches a j)arabola, meet one anotL•r, they will he cut in the same proportion. 42. If in a central conic parallel tangents he drawn at the extremities of a fixed diameter, and if hoth tangents be met hy any variable tangent, the rectangle under the intercepts on the parallel tangents is constant, being equal to the square on half the parallel dia- meter, i.e. the diameter conjugate to that joining the jwints of contact. 43. Any tangent to a hyperbola cuts off lengths from the asymp- totes whose product is constant. There is an obvious family likeness between these three consecu- tive propositions, and their arrangement in this manner can hardly liave been the result of mere accident. It is true that in. 42 [Prop. 66] is used almost directly afterwards for determining tlie foci of a central conic, and it might be supposed that it had its place in the book for this reason only; but, if this were the case, we should have expected that the propositions about the foci would follow directly after it instead of being separated from it by iii. 43, 44 [Props. 67, 68]. We have also a strong positive reason for supposing that the arrangement was due to set purpose rather than to chance, namely the fact that all three propositions can be used for describing a conic by means of tangents. Thus, if two tangents to a parabola are given, the first of the three propositions gives a general method of drawing CONSTRUCTION OF A CONIC BY MEANS OF TAXCFN'TS. cxxxl any number of other tangents ; while the second and tlnrd <,'ive tlie simplest cases of the construction of an ellipse and a hyperbjla by the same means, those cases, namely, in Λvhich the fixed tangents employed are chosen in a special manner. As therefore the three propositions taken together contain the essentials for the construction of all three conies by this method, it becomes important to inquire whether Apollonius possessed tlie means of drawing any number of tangents satisfying the given conditions in each case. That Apollonius was in a position to solve this problem is proved by the contents of two of his smaller treatises. One of these, λόγου άΐΓθτομη<; β" (two Books On cuttiwj of a proportion), we possess in a translation by Halley from the Arabic under the title De sectione rationis ; the other, now lost, was χωρίου άττοτομηζ β' (two Books On cutting off a space, which means cutting off from two fixed lines lengths, measured from fixed points on the lines respectively, such that they contain a rectangle of constant area). Now the very problem just mentioned of drawing any number of tangents to a parabola reduces precisely to that which is discussed with great fulness in the former of the two treatises, while the construction of any number of tangents to the ellipse and hyperbola in accordance with the conditions of III. 42, 43 [Props. 66, 67] reduces to two important cases of the general problem discussed in the second treatise. I. In the case of the parabola, if two tangents qP, qli and the points of contact P, R are given, we have to draw through any point a straight line which will intersect the given tangents (in r, J} respectively) in such a way that /*?• : rq = qp : pP, or Pr : Pq^qp :qR; that is, we must have Pr : qj) = Pq : qR (a constant ratio). In fact, we have to draw a line such that the intercept on one tangent measured from the point of contact is to the intercept on the other tangent measured from the intersection of the tangents in a given ratio. How to do this is shown in the greatest detail in the first Book λόγου άποτομη<;. If, again, instead of the points of contact, two other tjingents are given meeting the fixed tangent qP in r,, r^ and the fixed tangent qR in ;;,, p,^, we have to draw a straight line rp cutting off i •> CXXXU INTRODUCTION TO APOLLONIUS. along the tangents qP, qR parts measured from r,, jO, respectively which are in a given proportion, i.e. such tliat i\r : ρ J) = ?*,?•., : p^p„ (a fixed ratio) ; and this problem is solved in the second Book λογού άτΓοτομη<;. The general problem discussed in that treatise is, to draw from a point a straight line which shall cut off" from two given straight lines portions, measured from two fixed points A, B, which are in a given proportion, e.g., in the accompanying figure, OKM is to be drawn so that AM : BN is a given ratio. In the second Book of the treatise this general case is reduced to a more special one in which the fixed point Β occupies a position B' on the first line ΑΛί, so that one of the intercepts is measured from the intersection of the two lines. Tlie reduction is made by joining OB and drawing B'N' parallel to Β Ν from the point B' in which OB, MA intersect. Then clearly B'N' : BN is a given ratio, and therefore the ratio B'N' : AM is given. We have now to draw a straight line ON' Μ cutting MAB', B'N' in points J/", N' such that B'N' . . ^ =- a, given ratio, λ suppose. This problem is solved in the first Book, and the solution is substantially as follows. Draw OC parallel to N' B' meeting MA produced in C. Now suppose a point D found on AM such that CONSTRUCTION OF A CONIC BY MEANS OF TANCENTS. cxxxill Then, supposing that the ratio ' is niado ΐΜριαΙ to λ, we have AM B'N ' IV Μ AT) ~ OC ~ CM ' Ml) CB' AD~ CM' whence and therefore CM . MD = AD . CB' (a given rectangl••). Thus a given line CD has to be divided at Μ so that CM . MD has a given value ; and this is the Euclidean problem of applying to a given straight line a rectangle equal to a given area hut falling short, or exceeding, by a square. In the absence of algebraical signs, it was of course necessary for Apollonius to investigate a lai-ge number of separate cases, and also to find the limiting conditions of possibility and the number of the possible solutions between each set of limits. In the case repre- sented in the above figure the solution is always possible for any value of the given ratio, because the given value AD . CB', to which CM . MD is to be equal, is always less than CA . AD, and therefore (CD\^ -^ j , the maximum value of the rectangle whose sides are together equal to CD. As the application of the rectangle would give two positions of M, it remains to be proved that only one of them falls on ^Z> and so gives a solution such as the figure requires; and this is so because CM.MD must be less than CA . AD. The application to the parabola has more significance in the cases where the given ratio must be subject to certain limits in order that the solution of the problem may be possible. This will be so, e.g. in the annexed figure, where the letters have the same meaning as before, and the particular case is taken in which one .CAi.|FOKNlAL CXXXIV INTRODUCTION TO APOLLONIUS. intercept B'N' is measured from B', the intersection of the two fixed lines. Apollonius begins by stating the limiting case, saying that we obtain a solution in a special manner in the case where Μ is the middle point of CD, so that the given rectangle GM.MD or CB'.AD has its maximum value. In order to find the corresponding limiting value of λ, Apollonius seeks the corresponding position of D. „, , B'C CM B'M We have MD=AD=MA^ whence, since MD — CM, B'C _ C}1 B'M WIr'M~A'"WA' and therefore B' M"" = B'C.B'A. Thus Μ is determined, and therefore D also. According, therefore, as λ is less or greater than the particular OC value of _ thus determined, Apollonius finds no solution or two solutions. At the end we find also the following further determination of the limiting value of λ. We have AD = B'A+ B'C - (B'D + B'C) = B'A + B'C - 2B'M = B'A + B'C-2 J B'A . B'C. Thus, if we refer the various points to a system of coordinates with B'A, B'N' as axes, and if Ave denote the coordinates of by [x, y) and the length B'A by li, we have ■^J) h + x-2>Jhx' If we suppose Apollonius to have used these results for the parabola, he cannot have failed to observe that the limiting case described is that in which is on the parabola, while iV'OM is the tangent at ; for, as above, B'M _ B^ B'A ~ B'M = -ψ^, by parallels, CONSTRUCTION OF A CONIC BY MEANS OF TANGENTS. CXXXV SO that B'A, N'M are divided at J/, respectively in tlie same proportion. Further, if we put for λ the proportion between the lengths of the two fixed tangents, we obtain, if Λ, k be those lengths, k^ y /i h + x-2s/hx' which is the equation of the parabola referred to the fixed tangents as coordinate axes, and which can easily be reduced to the sym- metrical form 7/Ni ©'^( *; '• II. In the case of the ellipse and hyperbola the problem is to draw through a given point a straight line cutting two straight lines in such a way that the intei'cepts upon them measured from fixed points contain a rectangle of constant area, and for the ellipse the straight lines are parallel, while for the hyperbola they meet in a point and the intercepts on each are measured from the point of their intersection. These are particular cases of the general problem which, accord- ing to Pappus, was discussed in the treatise entitled χωρίον άττοτυμη ; and, as we are told that the propositions in this work corresponded severally to those in the λόγου άττοτομή, we know that the particular cases ηοΛν in question were included. We can also form an idea how the general problem was solved. The reduction to the particular case where one of the points from which the intercepts are measured is the intersection of the two fixed lines is effected in the same manner as in the case of proportional section described above. Then, using the same figure (p. cxxxii), we should take the point D (in the position represented by (Ό) in the figure) such that OC . AD =^ the given rectangle. We have then to draw the line ON'M so that B'N' .AM^OC .AD, B'N' AD UcT-AJr But, since B'N', OC are parallel, B'N' _ B'AI ~0C ~ CM' rx,. . -4 J/ AD DM Therefore CM= B'M^ BC' CXXXVi INTRODUCTION TO APOLLONIUS. and the rectangle B'M . MD = AD . B'C, which is given. Hence, as before, the problem is reduced to an application of a rectangle in the well-known manner. The complete treatment of the particular cases of the problem, with their διορισ/χοι, could present no difficulty to Apollonius. III. It is not a very great step from what we find in Apollonius to the general theorem that, if a straight line cuts off from tivo fixed straight lines intercepts, measured from given points on the lines respectively, which contain a rectangle of given area, the envelope of the first straight line is a conic section touching the two fixed straight lines. Thus, suppose Λ BCD to be a parallelogram described about a conic and E, F to be the points of contact of ΛΒ, CD. If a fifth tangent MN cuts AB, CD in M, iV and AD, CB in P, Q respectively, we have, by the proposition of Apollonius, EA.FD = EM.FN. Therefore Ε A EM AM AP FJ^~ YD~ Nb~ PD Hence, since Ε A - CF, CF FN CN AP~ PD~ AD' and therefore AP.CN=CF.AD, or the rectangle AP . CN has an area independent of the position of the particular fifth tangent MN. CONSTRUCTION OF A CONIC 15Y MEANS OF TAN(;ENTS. cxxxvil Conversely, if the lines AD, DC are given as well as the points A, C and the area of the rectangle AP . CN, we can deternune the point F, and therefore also the point Ε where Ali touches the conic. We have then the diameter EF and the direction of the chords bisected by it, as well as the tangent AD ; thus we can find the ordinate to EF drawn through the point of contact of AD, and hence we can obtain the equation of the conic referred to the diameter EF and its conjugate as axes of coordinates. Cf. Lemma XXV. of the first Book of Newton's Principia and the succeeding investigations. CHAPTER V. THE THREE-LINE AND FOUR-LINE LOCUS. The so-called τόπος (ττΐ τρβΓς καΙ τεσσάρας -γραμμας is, as Λνβ have seen, specially mentioned in the first preface of Apollonius as a subject Avhich up to his time had not received full treatment. He says that he found that Euclid had not worked out the synthesis of the locus, but only some part of it, and that not successfully, adding that in fact the complete theory of it could not be established Avithout the " new and I'emarkable theorems " discovered by himself and contained in the third book of his Conies. The words used indicate clearly that Apollonius did himself possess a complete solution of the problem of the four-line locus, and the remarks of Pappus on the subject (quoted above, p. xxxi, xxxii), though not friendly to Apollonius, confirm the same inference. We must further assume that the key to Apollonius' solution is to be found in the third Book, and it is therefore necessary to examine the propositions iu that Book for indications of the way in which he went to work. Tlie three-line locus need not detain us long, because it is really a particular case of the four-line locus. But we have, in fact, in in. 53-56 [Props. 74-76] what amounts to a complete demonstration of the theoretical converse of the three-line locus, viz. the proposition that, if from any point of a conic there he drawn three straight lines in fixed directions to meet respectively two fixed tangents to the conic and their chord of contact, the ratio of the rectangle contained by the first ttoo lines so drawn to the square on the third line is constant. The proof of this for the case where the two tangents are parallel is o])tained from iii. 53 [Prop. 74j, and the remaining three propo- sitions, iiL 54-56 [Props. 75, 76], give the proof where the tangents are not parallel. THE THREE-LINE AND FOUR-LINE LOCUS. CXXxix Tn like manner, we should expect to find the theorem of the four-line locus appearing, if at all, in the fornj of the converse proposition stating that every conic section has, tvith reference to any inscribed quadrilateral, the properties of the four-line locus. It will be seen from the note following Props. 75, 76 that this theorem is easily obtained from that of the three-line locus as presented by Apollonius in those propositions ; but there is nowhere in the Book any proposition more directly leading to the former. The explana- tion may be that the constriiction of the locus, that is, the aspect of the question which would be appropriate to a work on solid loci rather than one on conies, was considered to be of preponderant im- portance, and that the theoretical converse was regarded as a mere appendage to it. But, from the nature of the case, that converse must presumably have appeared as an intermediate step in the inΛ■estigation of the locus, and it could hardly have been unknown even to earlier geometers, such as Euclitl and Aristaeus, who had studied the subject thoroughly. In these circumstances we have to seek for indications of the probable course followed by Greek geometers in their investiga- tion of the four-line locus ; and, in doing so, we have to bear in mind that the problem must have been capable of partial solution before the time of Apollonius, and that it could be completely solved by means of the propositions in his third Book. We observe, in the first place, that iii. 54-56 [Props. 75, 76], which lead to the property of the three-line locus, are proved by means of the proposition that the ratio of the rectangles under the segments of any intersecting chords drawn in fixed directions is constant. Also the property of the three-line locus is a particular case of the property of a conic with reference to an inscribed quadri- lateral having t\vo of its sides parallel, that case, namely, in which the two parallel sides are coincident ; and it will be seen that the proposition relating to the rectangles under the segments of in- tersecting chords can equally well be used for proving generally that a conic is a four-line locus with reference to any inscribed quadrilateral which has two sides parallel. For, if A Β is a fixed chord of a conic and Jir a cliord in a given direction cutting ΛΒ in /, we have Rl.Ir . ^. ΖΓ77^ = (^""^'•)• If we measure JiK along Er equal to //•, the locus of A' is a chord Cxl LNTRODUCTION TO APOLLONIUS. DC meeting the diameter which bisects chords parallel to Rr in the same point in which it is met by ΛΒ, and the points D, C lie on lines drawn through A, JJ respectively parallel to Jir. Then, if x, y, z, u be the distances of R from the sides of the quadrilateral ABCD, we shall have — = (const.). yu ^ And, since A BCD may be any inscribed quadrilateral with two sides parallel, or a trapezium, the proposition is proved generally for the particular kind of quadrilateral. If Λve have, on the other hand, to find the geometrical locus of a point R Λvhose distances x, y, z, u from the sides of such a trapezium are connected by the above relation, we can first manipulate the constants so as to allow the distances to be measured in the directions indicated in the figure, and we shall have RI.RK R I .Ir ΎΓΤΐΒ ~ ΑΠΤΒ' where λ is a given constant. We must then try to find a conic whose points R satisfy the given relation, but we must take care to determine it in such a manner as to show synthetically at the same time that the points of the conic so found do really satisfy the given condition ; for, of cour.se, Λνβ are not yet supposed to know that the locus is a conic. It seems clear, as shown by Zeuthen, that the defective state of knowledge which prevented the predecessors of Apollonius from completing the determination of the four-line locus had reference rather to this first step of finding the locus in the particular case of a trapezium than to the transition from the case of a trapezium to that of a quadrilateral of any form. The transition was in fact, in THE TIIREE-LIXE AND FOrU-LINE LOCUS. Cxli itself, possible by means which won» within the cf)nipetence of Euclid, as will presently be seen ; but the ditiiculty in the way of the earlier step was apparently due to the fact that the conception of the two branches of a hyperbola as a sinjj;le curve had not occurred to any one before Apollonius. His preilecessors ac- cordingly, in the case \vhere the four-line locus is a complete hyperbola in the modern sense, probably considered only one branch of it ; and the question which branch it would be would depend on some further condition determining it as one of the two branches, e.g. the constant niiglit have been determined by means of a given point through which the conic or single-branch hyperbola, which it was required to prove to be the four-line locus, should pass. To pro\e that such a single branch of a hyperbola, not passing through all four corners of the quadrilateral, could be the four-line locus, and also to determine the locus corresponding to the value of λ leading to such a hyperbola, it was necessary to know of the connexion of one branch with the other, and the corresponding extensions of all the propositions used in the proof of the property of the inscribed quadrilateral, as well as of the various steps in the converse procedure for determining the locus. These extensions to the case of the complete hyperbola may, as already mentioned (p. Ixxxiv seqq.), be regarded as due to Apollonius. His predeces- sors could perfectly well have proved the proposition of the in- scribed trapezium for any single-branch conic ; and it will be seen that the converse, the construction of the locus, would in the particular case present no difficulty to them. The difficulty would come in where the conic was a hyperbola with two branches. Assuming, then, that the property of the four-line locus was established with respect to an inscribed trapezium by means of the proposition that the rectangles under the segments of intersecting chord.s are to one another in the ratio of the squares on the parallel tangents, what was wanted to complete the theory \vas (1) the extension to the case where the tangents are tangents to op- posite branches of a hyperbola, (2) the expression of the constant ratio between the rectangles referred to in tliose cases where no tangent can be drawn parallel to either of the chords, or where a tangent can be drawn parallel to one of them only. Now we find (1) that Apollonius proves the propo-sition for the case where the tangents touch opposite branches in in. 19 [Prop. 59, Case i.]. Also (2) the proposition in. 23 [Prop. 59, Case iv.] proves that, cxlii INTRODUCTION TO APOLLONIUS. where there is no tangent to the hyperbola parallel to either of the chords, the constant ratio of the rectangles is equal to the ratio of the squares of the parallel tangents to the conjtigate hyperbola ; and III. 21 [Prop. 59, Case ii.] deals with the case where a tangent can be drawn parallel to one of the chords, while no tangent can be drawn parallel to the other, and proves that, if tQ, the tangent, meets the diameter bisecting the chord to which it is not parallel in t, and if tq is half the chord through t parallel to the same chord, the constant ratio is then tQ^:tq'. Zeuthen suggests (p. 140) that the method adopted for deter- mining the complete conic described about a given trapezium ABCD, which is the locus with respect to the four sides of the trapezium corresponding to a given value of the constant ratio λ, may have been to employ an auxiliary figure for the purpose of constructing a conic similar to that required to be found, or rather of finding the form of certain rectilineal figures connected Avith such a similar conic. This procedure is exemplified in Apollonius, ii. 50-53 [Props. 50-52], Avhere a certain figure is determined by means of a previous construction of another figure of the same form ; and the suggestion that the same procedure was employed in this case has the advantage that it can be successfully applied to each of the separate cases in Avhich Apollonius gives the different expressions for the constant ratio between the rectangles under the segments of intersecting chords in fixed directions. We have the following data for determining the form of the conic similar to the required conic circumscribing ABCD : the value Hi Ir (λ) of the ratio -rj-jn between the products of segments of lines in two different directions, and the direction of the diameter P]) bisecting chords in one of the given directions. I. Suppose that the conic has tangents in both given directions (which is always the case if the conic is a conic in the old sense of the term, i.e. if the double-branch hyperbola is excluded). Let the points of the auxiliary figure be denoted by accented letters corresponding to those in the figure on p. cxl. We know the ratio and, if we choose any straight line for 0' l'\ we know (1) the position THE THREE-LINE AND FOUR-LINE L lOCUS. of a diameter, (2) its extremity P\ (3) the direction of the chords bisected by the diameter, (4) a point Q' with the tangent at that point. Then the intersection of the tangent at Q' with the diameter and the foot of the ordinate to it from Q' determine, with P\ three points out of four which are harmonically related, so that the remaining one, the other extremity {})') of the diameter, is found. Hence the conic in the auxiliary tigure is determined. II. Suppose that the conic has no tangent in either direction. In this case we know the ratio between the tangents to the hyperbola conjugate to the required auxiliary hyperbola, and Λve can therefore determine the conjugate hyperbola in the manner just described ; then, by means of the conjugate, the required auxiliary hyperbola is determined. III. Suppose that the conic has a tangent in the direction of AD, but not in the direction of ΛΒ. In this case, if the tangent Pt parallel to AD and the diameter bi.secting A Β meet in t, Apollonius has expressed the constant λ as the ratio between the squares of the tangent tP and of tq, the half of the chord through t parallel to AB. We have then tq tif If we now choose t'P' aibitrarily, we have, towards doterniiniiig the auxiliary similar conic, (1) a diameter with the direction of chords bisected by it, (2) one extremity P' of that diameter, (3) two points q, »' on the curve. cxli INTRODUCTION TO Al'OLLONIUS. If y^i Vi ''^''6 *^e ordinates of q^ s with respect to the diameter, .-Tj, x^ the distances of the feet of the ordinates from P', and .r/, a•/ their distances from the other (unkiiown) extremity of the diameter, we have is determined. The point ρ can thus be found by means of the ratio between its distances from two known points on the straight line on which it must lie. IV. Suppose that the conic has a tangent in tlie direction of AB, but not in the direction of AD. Let the tangent at P, parallel to AB, meet the diameter bisecting BC, AD vat, and let tq parallel to AD meet the conic in q ; we then have t'q t'F' If we choose either t'q or t'P' arbitrarily, we have (1) the diameter t'T', (2) the points P', q on the curve, the ordinates from which to the diameter meet it in t', T' respectively, (3) the tangent at P'. Since t'P' is the tangent at P', C't' .C'T' = \.a", where C is the centre, and a' the length of the diameter. THE THREE-LINE AND FOUR-LINE LOCUS. cxlv Therefore, by symmetry, T'q is the tangent at q. [Prop. 42.] Hence we can find the centre C by joining Γ', the middle point of Pq, to 0\ the point of intersection of the tangents, since Y'O' must be a diameter and therefore meets t'T' in C . Thus the auxiliary conic can be readily determined. The relation between the diameter a and the diameter h' conjugate to it is given by tig* _ δ;^ _ δ* σα . tr ~ a* ~ a' • Thus it is seen that, in all four cases, the propositions of Apollo- nius supply means for determining an auxiliary figure similar to that which is sought. The transition to the latter can then be made in various Avays ; e.g. the auxiliary figure gives at once the direction of the diameter bisecting AB, so that the centre is given; and we can effect the transition by means of the ratio between CA and CA'. There are, hoΛvever, indications that the auxiliary figures would not in practice be used beyond the point at which the ratio of the diameter (a) bisecting the parallel sides of the trapezium to its conjugate (ό) is determined, inasmuch as we find in Apollonius propositions which lead dii-ectly to the determination of the absolute values of a and b when the ratio j-(= -,-, ) is given. The problem to be solved is, in fact, to describe a conic through two given points A and Β such that one diameter of it lies along a given straight line, while the direction of the chords bisected by the diameter is given, iis well as the ratio (jj between the length of the diameter and its conjugate. Suppose that, in the accompanying figure, a straight line is drawn through Β parallel to the known direction of the diameter, H. C. cxlvi INTRODUCTION TO APOLLONIUS. and meeting DA produced in 0. Also let OB meet the curve (which Λve will suppose to be an ellipse) again in E. Then we must have OB.OE a? OA.OD'b" whence OE can be found, and therefore the position of E. The line bisecting BE and parallel to ylZ> or BC will determine the centre. AVe have now, for the case of the ellipse, a proposition given by Apollonius which determines the value of a* directly. By III. 27 [Prop. 61 (1)] we know that OB' + OE' + ^; {0Λ' + OD') = a\ whence a' is at once found. Similar propositions are given for the hyperbola (see ill. 24-26, 28, 29 [Props. 60 and 61 (2)]). The construction in the case of the hyperbola is also facilitated by means of the asymptote properties. In this case, if the letters have the same significations as in the figure for the ellipse, we find the centre by means of the chord BE or by using the auxiliary similar figure. The asymptotes are then determined by the ratio γ. If these cut the chord AD in K, Z, then ΑΚ.ΑΙ = ψ, or AK.KD=lh\ If the required curve is a parabola, the determination of the auxiliary similar figure after the manner of the first of the four cases detailed above would show that P', the end of the diameter, is at the middle point of the intercept between the intersection of the diameter with the tangent at Q' and with the ordinate from Q' i-espec- tively. The curve can then be determined by the simple use of the ordinary equation of the parabola. So far the determination of the four-line locus has only been considered in the particular case Avhere two opposite sides of the inscribed quadrilateral are parallel. It remains to consider the possible means by which the determination of the locus with reference to a quadrilateral of any form whatever might have been reduced to the problem of finding the locus with reference to a trapezium. As Apollonius' third Book contains no propositions which can well be used for effecting the transition, it must be \ THE THREE-LINE AXD FOUR-LINE LOCUS. cxlvii concluded that the transition itself was not affected by Apollonius' completion of the theory of the locus, but that the key must be looked for elsewhere. Zeuthen (Chapter 8) finds the key in the Poi'isms of Euclid*. He notes first tliat Archimedes' proposi- tion (given on p. lix, Ix above) respecting the parabola exhibits the curve as a four-line locus with respect to two quadrilaterals, of which one is obtained from the other by turning two adjacent sides about the points on the parabola in which they meet the two other sides. (Thus PQ is turned about Q and takes the position QT, while PF is turned about its intersection with tlie parabola at infinity and takes the position of the diameter through Q.) This suggests the inquiry whether the same means >vliich are used to effect the transition in this very special case cannot also be employed in the more general case now under consi- deration. As the Porisms of Euclid are themselves lost, it is necessary to resort to the account Λvhicll Pappus gives of their contents ; and the only one of the Porisms which is there preserved in its original form is as follows t : If from tivo given points there be drmvn straight Hues which intersect one another on a straight line given in position, and if one of the straight lines so dra^vn cuts off from a straight line given in position a certain length measured from a given point on it, then the other straight line also tvill cut off a portion from another straight line hearing a given ratio [to the former intercept^ The same proposition is true also when a four-line locus is substituted for the first-mentioned given straight line and the two fixed points are any two fixed points on the locus. Suppose that we take as the two fixed points the points A and C, being two opposite corners of the quadrilateral A BCD to which the locus is referred, and suppose the lines from which the intercepts are cut off to be CE, A Ε drawn respectively parallel to the sides Β A, EC of the quadrilateral. Let Μ be a point on the required locus, and let AD, J J/ meet * That the Porisim of Euclid were a very important contribution to geometry is indicated by the description of them in Pappus (p. G48) as a collection most inRenionsly adapted for the solution of the more weighty problems (άθροιαμ-α φιΚοτΐχνότατον (is την άνά\νσιι> των ϊμβρίθίστέρων ττροβΧηματων). t Pappus, p. ϋόΟ. cxlviii INTRODUCTION TO APOLLONIUS. CE in D', M' respectively, while CD, CM meet ΑΣ in D", M" respectively. For the purpose of determining the geometrical locus, let the distances of Μ from ΛΒ, CD be measured parallel to BC, and its distances from BC, AD parallel to ΒΛ. Then the ratio of the distances of Μ from CD, BC respectively Λνϋΐ be equal to ^^ — , and the ratio of the distances of J/ from AB, Li hd AE DA will be equal to -fttt?, • ^ D 31 Therefore the fact that the ratio of the rectangles under the distances of Μ from each pair of opposite sides of the quadrilateral A BCD is constant may be expressed by the equation D"3r . CE .,. -mf^^AE = ''^ say (1), where /i is a new constant independent of the position of M. If now λ be determined by means of the position of a point F of the locus, we have D"M" _ D"F" _ F"M " D'M' " D'F' " F'M' ^*'^' where F\ F" are the intersections of AF, CE and of CF, AE respectively. And, since the last ratio in (2), which is derived from the other two, remains constant while Μ moves along the required locus, it follows that that locus is also a four-line locus with reference to the four sides of the quadrilateral ABCF. Thus, in order to extend the proposition about an inscribed THE THREE-LINE AND FOUR-LINE LOCUS. cxlix trapezium to a quadrilateral of any forra, or, conversely, to reduce the determination of a four-line locus with reference to any quadri- lateral to a similar locus with reference to a trapezium, it was only necessary to consider the case in which one of the lines AD or AF coincides with AF. It follows that the four-line locus with reference to any quadrilateral is, like the four-line locus with reference to a trapezium, a conic section. The actual determination of the locus in the general form can be effected by expressing it in the more particular form. Suppose that the distances of Μ from AB, CD (reckoned parallel to BC) are denoted by x, z, and the distances of Μ from BC, A D (reckoned parallel to Β A) are y, u respectively. Then the locus is determined by an equation of the form xz = \.yii (1), where λ is a constant, and x, y are the coordinates of the point Μ Avith reference to BC, Β A as axes. If /*, Q are the points in which the ordinate (y) of Μ meets A D, ΛΕ respectively, u = PM = PQ-MQ (2). Since (— MQ) is the distance of Μ from A Ε measured parallel to Β A, let it be denoted by w, . Then, from the figure, Therefore, from (1), z — \ , y ) , we derive from the figure Ώ"Μ" '=-cjr'y^ and we have then to take a point G on AE such that D'E _D"G AE ~ CE ' (The point G is thus seen to be a point on the locus.) cl INTRODUCTION TO APOLLONIUS. , D'E D"M" D"G Heuce ^-^ae'-^^ CE '^ ' CE'J GM" ~ GE -y where «, is the distance of the point Μ from the line CG measured parallel to BC. The equation representing the locus is accordingly transformed into the equation xz^ = λ . 2/w, , and the locus is expressed as a four-line locus with reference to the trapezium ABCG. The method here given contains nothing which would be beyond the means at the disposal of the Greek geometers except the mere notation and the single use of the negati\^e sign in (- 3iQ), which however is not an essential difference, but only means that, whereas by the use of the negative sign we can combine several cases into one, the Greeks would be compelled to treat each separately. Lastly, it should be observed that the four-line locus with reference to a trapezium corresponds to the equation ax' + βχι/ Λ- yy' 4- dx + e7j = 0, which may be written in the form X (ax + fiy + d) = -y {yy + e). Thus the exact determination of the four-line locus with reference to a trapezium is the problem corresponding to that of tracing a conic from the general equation of the second degree wanting only the constant term. CHAPTER VI. THE CONSTRUCTION OF A CONIC THROUGH FIVE POINTS. Since Apollonius was in possession of a complete solution of the problem of constructing the four-line locus referred to the sides of a quadrilateral of any form, it is clear that he had in fact solved the problem of constructing a conic through five points. For, given the quadrilateral to Λvhich the four-line locus is referred, and given a fifth point, the ratio (λ) between the i-ectangles contained by the distances of any point on the locus from each pair of opposite sides of the quadrilateral measured in any fixed directions is also given. Hence the construction of the conic through the five points is reduced to the construction of the four-line locus where the constant ratio λ is given. The problem of the construction of a conic through five points is, however, not found in the work of Apollonius any more than the actual determination of the four-line locus. The omission of the latter is easily explained by the fact that, according to the author's own words, he only professed to give the theorems which were necessary for the solution, no doubt regarding the actual construc- tion as outside the scope of his treatise. But, as in Euclid we find the problem of describing a circle about a triangle, it would have been natural to give in a treatise on conies the construction of a conic through five points. The explanation of the omission may be that it was not found possible to present the general problem in a form sufficiently concise to be included in a treatise embracing the whole subject of conies. This may be easily understood when it is remembered that, in the first place, a Greek geometer would regard the problem as being in reality three problems and involving a separate construction for each of the three conies, the parabola, the ellipse, and the liyperbola. He would clii INTRODUCTION TO APOLLONIUS. then discover that the construction was not always possible for a parabola, since four points are sufficient to determine a parabola; and the construction of a parabola through four points would be a completely diflerent problem not solved along with the construction of the four-line locus. Further, if the curve were an ellipse or a hyperbola, it would be necessary to find a ^ωρισμόζ expressing the conditions Λνΐύΰΐι must be satisfied by the particular points in order that the conic might be the one or the other. If it were an ellipse, it might have been considered necessary to provide against its degeneration into a circle. Again, at all events until the time of ApoUonius, it would have been regarded as necessary to iind a διορισ /xos expressing the conditions for securing that the live points should not be distributed over both branches of the hyperbola. Thus it would follow that the complete treatment of the problem by the methods then in use must have involved a discussion of con- siderable length which Avould have been disproportionate in such a work as that of ApoUonius. It is interesting to note how far what we actually find in ApoUonius can be employed for the dii-ect construction of a conic through five points independently of the theory of the four-line locus. The methods of Book IV. on the number of points in Λvhich two conies may intersect are instructive in this connexion. These methods depend (1) on the harmonic polar property and (2) on the relation between the rectangles under the segments of intersecting chords drawn in fixed directions. The former property gives a method, Λvhen five points are given, of determining a sixth ; and by repeating the process over and over again we may obtain as many separate points on the curve as we please. The latter proposition has the additional advantage that it alloAvs us to choose more freely the particular points to be determined ; and by this method Λνο can find conjugate diameters and thence the axes. This is the method employed by Pappus in determining an ellipse passing through five points respecting Λvhich it is known beforehand that an ellipse can be drawn through them* It is to be noted that Pappus' solution is not given as an independent problem in conic sections, but it is an intermediate step in another problem, that of finding the dimen- sions of a cylinder of which only a broken fragment is given such that no portion of the circumference of either of its bases is left whole. Further, the solution is nmde to depend on what is to be * Pappus (ed. Hultsch), p. 107G seqq. THE CONSTRUCTION OF A CONIC TUllOUGH I'lVK I'oINTS. rliii found in ApoUonius, and no claim is advanced that it contains anything more than any capable geometer could readily deduce for himself from the materials available in the Conies. Pappus' construction is substantially as follows. If the "iven points are A, B, C, D, E, and are sucli that no two of the lines connecting the different pairs are parallel, we can reduce the problem to the construction of a conic through A, B, />, E, F, where EF is parallel to AB. For, if EF be drawn through Ε parallel to AB, and if CD meet AB in and EF in 0', we have, by the proposition relating to intersecting chords, CO.OD : AO. OB = CO' . O'D : EC . O'F, whence O'F is known, and therefore F is detoriiiined. We have therefore to construct an ellipse tli rough J, />', /), E, F, where EF is parallel to AB. And, if V, )V he the middle points of AB, EF respectively, the line joining V and W is a diameter. Suppose BB to be the chord through JJ parallel to the diameter, and let it meet AB, EF in G, U respectively. Then R is deter- mined by means of the relation RG.CD -.BG.GA -RlI.llD : FH .UK (1). cliv INTRODUCTION TO APOLLOXIUS. In order to detenuine R, let I) J}, RA be joined meeting EF in A", L respectively. Then RG . GD : BG . GA = {RH : IIL) . {DII : UK), l^y similar triangles, = RH.IID : Κ II. IIL. Therefore, from (1), we have FU.HE^KII.HL, whence IIL is found, and therefore L is determined. And the intersection of AL, DH determines R. In order to find the extremities of the diameter (PP'), we draw £D, RF meeting the diameter in M, Ν respectively. And, by the same procedure as before, Λve obtain /'//. HE : RII. II D = FW . WE : P'W . WP, by the property of the ellipse. x\lso FH . HE : RH .HIJ = F W . WE : iV W . WM, by similar triangles. Hence P' W . WP = Ν W . WM ; and similarly we can find the value of P'V. VP. Pappus' method of determining P, P' by means of the given A'alues of P' V . VP and FW . WP amounts to an elimination of one of the unknown points and the determination of the other by an equation of the second degree. Take two points Q, Q' on the diameter such that FV. VP= WV. VQ (a), P'W.WP= VW.WQ' (β), and V, W, Q, Q' are thus known, while P, P' remain to be found. It follows from (a) that FV : VW=QV: VP, whence FW -.VW^PQ: Ρ V. From this we obtain, by means of (/3), PQ .PV=Q'W : WP, so that PQ -.QV^Q'W-.PQ', or PQ.PQ'^QV.Q'W. Thus Ρ can be found, and siinihuly /''. THE CONSTUKCTION OF Λ CONIC TIllloUlMl FIVE I'OINTS. civ It is noteworthy that Pappus' method of determining the ex- tremities of the diameter PP' (which is the principal oVyect of his construction) can be applied to the direct construction of the points of intersection of a conic determined by five points with any straight line whatever, and there is no reason to doubt that this construction could have been effected by Apollonius. But there is a simpler expedient which we know from other sources that Apollonius was acquainted with, and Avhich can be employed for the same purpose when once it is known that tlie four-line locus is a conic. The auxiliary construction referred to formed the suljject of a whole separate treatise of Apollonius On deter inhtate section (ττΐρι 8ιωρισμ.€νηζ TOfxrj•;). The problem is as follows : Given four points A, B, C, D on a straight line, to det(irmine another point Ρ on the same straight line so that the ratio AP.CP-.BP. DP has a given value. The determination of the points of intersection of the given straight line and a four-line locus can be immediately transformetl into this problem. A, B, C, D being in fact the points of intei-section of the given straight line with the four lines to which the locus has reference. Hence it is important to examine all the evidence which we possess about the separate treatise referred to. This is contained in the seventh Book of Pappus, who gives a short account of the contents of the Avork* as well as a number of lemmas to the different propositions in it. It is clear that the question was very exhaustively discussed, and in fact at much greater length than would have been likely had the investigation not been intended as a means of solving other important problems. The conclusion is therefore irresistible that, like the Books λόγου απότομης and χωρίον άποτομη<; above mentioned, that On determinate section also was meant to be used for solving problems in conic sections. To determine Ρ by means of the equation AP.CP^X.BP.DP, where A, B, C, D, λ are given, is now an easy matter because the problem can at once be put into the form of a quadratic equation, and the Greeks also would have no difficulty in reducing it to the usual application of areas. But, if it was intended for application • Pappus, pp. 042 — 644. clvi INTRODUCTION TO APOLLONIUS. in further investigations, the complete discussion of it would naturally include, not only the finding of a solution, but also the determination of the limits of possibility and the number of possible solutions for ditierent positions of the given pairs of points A, C and B, D, for the cases where the points in either pair coincide, where one is infinitely distant, and so forth : so that we should expect the subject to occupy considerable space. And this agrees with what we find in Pappus, Λvho further makes it clear that, though we do not meet with any express mention of series of point-pairs deter- mined by the equation for different values of λ, yet the treatise contained what amounts to a complete theoi-y of Involution. Thus Pappus says that the separate cases were dealt with in which the given ratio was that of either (1) the square of one abscissa measured from the required point or (2) the rectangle contained by two such abscissae to any one of the following : (1) the square of one abscissa, (2) the rectangle contained by one abscissa and another separate line of given length independent of the position of the required point, (3) the rectangle contained by two abscissae. We also learn that maxima and minima wei-e investigated. From the lemmas too we may draw other conclusions, e.g. (1) that, in the case Avhere λ=1, and therefore Ρ has to be determined by the equation AP.CF = BP.DP, Apollonius used the relation* BP :DP = AB.BG: AD . DC ; (2) that Apollonius probably obtained a double point Ε of the involution determined by the point-pairs A, C and B, D by means of the relation t AB . BG ■.AD.DC = BE' : DE\ Assuming then that the results of the work On determinate section were used for finding the points of intersection of a straight line with a conic section represented as a four-line locus, or a conic determined by five points on it, the special cases and the A'arious Χωρισμοί would lead to the same number of properties of the conies under consideration. There is therefore nothing violent in the supposition that Apollonius had already set up many landmarks in the field explored eighteen centuries later by Desargues. • This appears in the first lemma (p. 704) and is proved by Pappus for several different cases. t Cf. Pappus' prop. 4U (p. 732). APPENDIX TO INTRODUCTION. NOTES ON THE TERMINOLOGY OF C4REEK GEOMETRY. The propositions from the Conies of Apollonius which are given at length in Chapter II. above will have served to convey some idea of the phraseology of the Greek geometers ; and the object of the following notes is to supplement what may be learnt from those propositions by setting out in detail the principal technical terms and expressions, with special reference to those which are found in Apollonius. It will be convenient to group them under different headings. 1. Points and lines. A point is σημίίον, the point A to A σημΐΐον or το A simply ; a fuller expression commonly used by the earlier geometers was to (σημΐΐον) ίφ' ου A, "the point on which (is put the letter) A*." Any point is τνχον σημίΐον, the j^oint (so) arising το yci'o'/xcvov (τημίΐον, tlie point (so) taken το ληφθέν σημάον, a point not ivithin the section (τημάον μη Ιντος της τομής, any point within the surface σημίΐόν τι των cvTos τ^5 €7Γΐφαν€ΐα9 ; in one point only καθ' tv μόνον σημ€ΐον, in two points κατά δυο, and so on. The following are names for particular points : apex or vertex κορνφη, centre κίντρον, point of division διαίρίσι?, ])oint of bisection 8ιχοτομία, extremity iripas. A line is γραμμή, a straight line €νθ(ΐα -γραμμή or €νθ€Ϊα alone, a finite straight line eWeia ττίπ^ρασμίνη ; a curved line is καμιήλη * A similar expression was ή {ΐύθύα.) ΐφ' rj .\B the gtniinht line {on which are the letters) AB. The same phrases, with the same variation of ca.'ie after txl, are found frequently in Aristotle, particularly in the logical trefttises and the Physics. civiii APPENDIX TO IXTRODrCTIOX. γραμμή, but γραμμή alone is ofton used of a curve, e.g. a circle or a conic ; thus το ττερας της ΐνθΐία<; το ττρο? rrj γραμμτ) is that extremity of the straight line ivhich is on the curve. A sec/ment (of a line as Avell as a curve) is τμήμα. Of lines in relation to other lines we find the terms parallel τταράλληλος, a peiyendicular to κάθίτο<ί ctti (with ace); a straight line jyroduced is 77 eV cv^cta? αύτ^. For a line passing through particular points Ave have the follow- ing expressions used with Sta and the genitive, r^^ii^ Ιρχεται, «λίυσίται, ΤΓορΐνΐταί ; likewise πίπτω δια, or κατά (with ace). Of a line meeting another line πίπτίΐν Ιπί (with ace), σνμπίπτΐΐν, συμβάλλΐΐν, άπτομαι are used ; until it meets is Iws ov σνμπίστι or a;(pis αν συμπέστ], point of meeting σνμπτωσι^ ; tlie line from the jjoint of concourse to Δ, η άπο τηζ σνμπτωσΐως ctti to Δ ; the straight line joining H, Θ, η cVi τα Η, Θ Ιπιζίνγννμίνη evOeLa ; ΒΑ passes through the points of contact, ΙττΧ τα? άφα'ς Ιστιν τ; ΒΑ. The line ΖΘ is bisected in Μ, δίχα τίτμ-ηται η ΖΘ κατά τό Μ ; bisecting one another διχα τίμνονσαι άλλι;λα?, the line joining their middle jioints η τάς Βιχοτομίας αντών ίπιζίνγννονσα, is cut into equal and unequal parts «is μ\ν ίσα, eis 8e άνισα τίτμηται. Straight lines cut off ov intercejyted are αποτ€/χνο'/α£ναι or άπολαμ- βανόμ€ναι, the part cut off ivithout (the curve) η cktos άπολαμβανομίνη, ivill cut off an eqrial length Ισην άποληφ€ται, the lengths intercepted on it bi/ the [conic) section totvards the asyynptotes at άπολαμβανόμΐναι aV αυτηζ προς ταις ασυ/Ατττωτοις. A point on a line is often elegantly denoted by an adjective agreeing with it : thus αττ' άκρα? αντηζ from its extremity, απ άκρου τον άξονος from the extremity of the axis, η eV ακραν την άποΧηφθ^,σαν αγομένη the line drawn to the extremity of the intercept, at προς μίσ-ην την τομην κλωμίναί (Ιθίΐαι the straight lines drawn so as to meet at the middle point of the section. 2. Angles. An angle is γωνία, an acute angle o^tia γωνία, obtuse α//,^λ€Γα, right όρθη ; at right angles to προς ορθάς (with dative) or ορθός προς (with ace); the line Δ A (drawn) from Δ at right angles to ΕΔ, από τοΰ Δ τ^ ΕΔ όρθη r; ΔΑ ; to cut at right angles προς ορθας Tt/Aveiv, tvill tiot in general be at right angles but only ivhen... ονκ a'ul προς ορθας ίσται, αλλ' όταν... NOTES ON THE TERMINOLOGY OF OREKK ΟΕοΜΚΤΙίΥ ■γωνία. Vei'ticalliJ opposite (angles) κατά κορνφην άλλί;λαις /cci/xcrut ; f/tf angle vertically opposite to the angle ΖΘΕ, η κατά κορνφψ τψ νπο ΖΘΕ γωνίας ; the same expression is also used of triangles (e.g. in τα •γινόμενα κατά κορνφην τρίγωνα), and of the two halves of a double cone, which are called vertically opposite surfaces αί κατά κορνφην €τηφάν€ΐαι. The exterior angle of the triangle is η Ικτο<; τον τρίγωνου γωνιά. For the angle ΔΓΕ we find the full expression η ττζραχομά'η -γωνία ντΓο των ΔΓΕ or "the angle contained by the lines ΔΓ, ΓΕ," but more usually η νπο των ΔΓΕ or η νττό ΔΓΕ. The angles ΑΓΖ, ΑΖΓ α?•(3 (together) equal to a right angle ai viro ΑΓΖ, ΑΖΓ μια. ορθή Γσαι €ίσιν. The adjacent angle, or the sjipplement of an angle, is η €φ€ξη<; -γωνία. To subtend (an angle) is νττοτζίναν either Avith a simple accusa- tive, or with νπο and ace. (extend under) as in at γωνι'αι, ΰφ' άς αί ομόΧογοι ττλίυραι νποτύνονσιν the angles which the homologous sides stibtend. 3. Planes and plane figures. A phne is eVtTreSov, a figure σχήμα or €1809, a figure in the sense of a diagram καταγραφή or σχήμα. (A circle) which is not in tlie same ])lane icith the point 05 oJk Ιστιν €V τω αυτω ΙπιττίΒω τω σημ^ίω. The line of intersection of two planes is their κοινή το/χτ;. A rectilineal fig^ire is σχήμα ΐνθνγραμμον (Euclid), and among the figures of this kind are triangle τρίγωνον, quadrilateral τΐτράπλενρον, a five-sided figure πίντάπλ€νρον etc., ττλευρά being a side. A circle is κύκλος, its circximference ττίριφίρίΐα, a semicircle ημικνκλιον, a segment of a circle τ/Αΐ7/χα κνκλου, a segment greater, or less, than a semicircle τμήμα μύζον, or έλασσον, ημικνκΧίον ; a segment of a circle containing an angle equal to tlie angle ΑΓΒ is κνκ\ον τμήμα Ζνχόμΐνον γωνίαν ΐσην ttj νττό ΑΓΒ. Of quadrilaterals, a parallelogram is παραλληλόγραμμον, a square τίτρα'γωνον, a rectangle όρθογώνίον or frequently χωρίον with ur without ο'ρ^ογωνιον. Diagonal is δια/χ€τρος. To describe a figure upon a given line (as base) is ανάγραφαν από. Thus the figure ΘΙΗ has been described upon tlie radius ΘΗ is άναγί- γραπται άπυ τη<ϊ «κ τοΰ κέντρου τη<: ΘΗ ίΓδος το ΘΠΙ, the square υη ZW clx APPENDIX TO INTRODUCTION. is TO άτΓο τηζ ΖΘ (τίτροίγωνον), the figures on ΚΛ, ΛΖ is τά άπο ΚΛΖ cffi?/. But Ιπί with the genitive is used of describing a semicircle, or a segment of a circle, on a given straight line, e.g. cVt ^75 ΑΔ γίγραφ^ω ημίκύκλων, τρ.ΐ7/χα κύκλου. Similarly quadrilaterals standing on the diameters as bases are βίβηκότα inl τών διαρ,€'τρων τ£τραπλ€υρα. A rectangle applied to a given straight line is ΐΓαρακίίμ.(νον -παρά (with ace), and its breadth is ττλάτο?. The rectangle contained by ΔΖ, ZE is TO υτΓο των ΔΖ, ZE or το νπο {των) ΔΖΕ ; imll contain (with another straight line) a rectangle equal to the sqtiare on is taov ΤΓζρύξίΐ τω από. With reference to squares the most important point to notice is the use of the word SiW/Ats and the various parts of the verb StVa/iiai. δυνα/χις expresses a square (literally a potver) ; thus in Diophantus it is used throughout as the technical term for the square of the unknown in an algebraical equation, i.e. for af. In geometrical language it is used most commonly in the dative singular, Βυνάμει, in such expressions as the following : λόγο? oV Ιχίΐ τό εντός τ/Αΐ7/χα ττρος το λαπον 8ννάμ€ΐ, " the ratio which " (as one might say) " the inner segment has to the remaining segment j)ote7itially," meaning the ratio of the square of the inner segment to that of tlie other. (Similarly Archimedes speaks of the radius of a circle as being Ζννάμίΐ Ισα to the sum of two areas, meaning that the square of the radius is eq^ial etc.) In like manner, when δύναται is used of a straight line, it means literally that the line is (if squared) capable of producing an area equal to another, ίσον δυνοί/χεναι τω υπό is in Apollonius (straight lines) the squares on tvhich are equal to the rectangle contained by ; δύναται τό 7rcpic;(o/x.€vov νπο the square on it is equal to the rectangle contained by ; MN δύναται τό ΖΞ, the square on MN is equal to the rectangle Zs. ; Βννησίται τό παρακύμ^νον ορθο-γωνων ττρός Τ7;ν προσπο- ρισθάσαν the square on it will be equal to the rectangle applied to the straight line so taken in addition (to the figure) ; and so on. To construct a triangle out of three straight lines is in Euclid L• τριών €ΐ5^€ΐών τρίγωνον συστϊ^σασβαι, and similarly Apollonius speaks of its being possible σνστησασθαι τρίγωνον ck τη<; Θ και δυο τών ΕΑ, to construct a triangle from the straight line Θ and ttco straight lines {equal to) EA. The triangle formed by three straight lines is τό νινό /xcvov υτΓ αυτών τρίγωνον. Equiangular is ίσογω'νιος, similar όρ,οιο?, similar and similarly sitiiated ό/χοιος και ό/χοιω? kci/xcvos ; because of the similarity of the triangles ΘΕΝ, KEO is δια T17V ομοιότητα των ΘΕΝ, ΚΕΟ τριγώνων. NOTES ON THE TERMINOLOGY OF GREEK GEOMETRY. clxi 4. Cones and sections of cones. A cone is κώνο?, a right cone όρθος κώνο•;, an oblique or scalene cone σκαλην6<; κώνο?, the surface of a cone is κωνική (τηφάναα, the straiyht line generating the surface by its motion about the circumference of a circle is η γράφουσα εΰθ^Ια, the fixed jyoint through which the straight line always passes is το μ€μ£νηκ6ς σημ^ίον, the surface of the double cone is that ichich consists of tzvo surfaces lying vertically opposite to one another η σνγκΐΐται €κ 8vo ίπιφανίΐων κατά κορνφην άλλϊ/λαις καμένων, the circular base is βάσι<;, the apex κορυφή, the aans άξων. A circular section subcontrary to the base is υπεναντια τομή. In addition to the names parabola, ellipse, and hyperbola (which last means only one branch of a hyperbola), Apollonius uses the expression τομαΐ άντίκύμ^ναι or αί άντικύμ^ναι denoting the opposite branches of a hyperbola ; also at κατ evavTiov τομαί has the same meaning, and we even find the expression Βιάμ€τρο<; των 8vo σνζνγών for a diametei' of ttvo jmirs of opposite branches, so that conjugate here means opposite branches. (Cf. too ev μίν ttj Ιτίρα συζυγία in the one pair of opposites.) Generally, however, the expression τομαΙ σνζνγ€Ϊ<; is used of conjugate hyperbolas, which are also called αί κατά σνζυγίαν άντίκάμ^ναι or συζνγ(χ<; άντικύμίναι conjugate opposites. Of the four branches of two conjugate hyperbolas any two adjoining branches are αί Ιφ(.ξη<; τομαί. In the middle of a proposition, where we should generally use the word curve to denote the conic, Apollonius generally uses τομή sectimi, sometimes γραμμή. 5. Diameters and chords of conies. Diameter is ή δια/χίτρο?, conjttgate diameters συζυγής Βιάμΐτροι, of which the transverse is η ιτλαγία, the other η όρθια {erect) or Sorrcpa {secondary). The original diameter (i.e. that first arising out of the cutting of a cone in a certain manner) is η Ικ τη<; γενβ'σεω? 8ιάμ(τρο<; or η προϋ- πάρχουσα 8ιάμ€τρος, and (in the plural) αί αρχικά) 8ιάμ(τροι. The bisecting diameter is η διχοτομούσα 8ιάμ€τρος. A radi us of a central conic is simply €κ του κέντρου (with or without the definite article). Chords are simply αί άγόμ^ναι iv ttj τομτ}. 6. Ordinates. The word used is the adverb τίταγμίνω<; ordinate-wise, and the advantage of this is that it can be u.scd with any part of the verb H. C. ^ Cbdi APPENDIX TO INTRODUCTION. signifying to draw. This verb is either κοτάγειν or ανάγειν, the former being used when the ordinate is drawn doion to the diameter from a point on the curve, and the latter when it is drawn uptvards from a point on a diameter. Thus τίταγ/Αβ'νω? κατηχθω inl την διοί/χίτρον means suppose an ordinate drawn to the diameter, which diameter is then sometimes called -ή Ιή> ην άγονται or κατηκται. An ordinate is τεταγ/χενως καταγόμενη or κατη-γμίνη, and sometimes τ€ταγ- μ€νω<; alone or κατηγμίνη alone, the other word being understood; similarly κατηκται and άνηκται are used alone for is an ordinate or has been drawn ordinate-wise. τεταγμίνως is also used of the tangent at the extremity of a diameter. Parallel to an ordinate is τταρά τεταγμίνωζ κατηγμένην or τταρα- τεταγ/Αίνω? in one word. 7. Abscissa. The abscissa of an ordinate is η αποΧαμβανομίνη νπ αντης άπο της Βίαμετρον ττρος rfj κορνφτ) the (portion) cut off by it from the diameter towards the vertex. Similarly we find the expressions αί άποτΐμνόμεναι νττο της κατηγμενης, or αί άττολαμβανόμεναί νπ αυτών, ττρος τοις πέρασι της ττλαγίας ττλενρας τον ΐί8ους the ( portions) cut off by the ordinate, or by them, towards the extremities of the transverse side of the figure (as to which last expression see paragraph 9 following). 8. Parameter. The full phrase is the parameter of the ordinates, which is η παρ' ην δύνανται αί κατα-γόμεναι τεταγμίνως, i.e. the straight line to which are applied the rectangles which in each conic are equal to the squares on the ordinates, or (perhaps) to which the said squares are related (by comparison). 9. The "figure" of a central conic. The figiire (to cTSos) is the technical term for a rectangle supposed to be described on the transverse diameter as base and with altitude equal to the parameter or latus rectum. Its area is therefore equal to the square on the conjugate diameter, and, with reference to the rectangle, the transverse diameter is called the transverse side (πλαγία πλευρά) and the parameter is the ei-ect side (ορθία πλευρά) of the figure (εΤ8ος). We find the following different expressions, to προς tyj ΒΔ εΤΒος the figure on (the diameter) ΒΔ ; το τταρα την AB cT8o5 the figtire applied to (the diameter) AB ; το υπο ΔΕ, Η ct8os the figure contained by (the diameter) ΔΕ and (the parameter) NOTES ON THE TERMINOLOGY OF fillEEK OEOMETUY. clxiU H. Similarly to γινόμ€νον ίΤδον ττρό? τη δια tt7S άφη•; ayo/ACVTj διαμίτρψ is the figure formed on the diameter drawn through the point of coiitact and TO ττρο^ rrj τά? άφας ίπιζ€νγνυονσΎ) ctSos is the figure on [the diameter which is) the chord joining the points of contact (of two parallel tangents). TO τέταρτον του «Γδους onefourth of the figure is, with reference to a diameter PP', one-fourth of the square of the conjugate diameter DD\ i.e. CD-. 10. Tangents etc. To touch is most conniiouly ίφάπτίσθαι, whether used of straight lines touching curves or of curves touching each other, a tangent being of course Ιφαπτομίνη ; the tangent at Λ, ή κατά. το Λ εφαπτομένη. (The simple verb απτΐσθαι is not generally used in this sense but as a rule means to meet, or is used of points Ii/ing on a locus. Cf. Pappus, p. 664, 28, άψεταί τό σ-ημίίον θέσ^ι δβδο/χενης ίνθ(.ία% the point tvill lie on a straight line given in position ; p. 664, 2, lav άιττηται eVi- 7Γ€δου τόπου θέσ^ί δβδο/χενου if it lies on a plane locus given in position). The word iwnj/aveiv is also commonly used of touching, e.g. καθ' eV ίτΓΐι/^αυ'ουσα τη^ τομής is touching the section in one point, ης έτνχΐ των τομών innj/avovaa toucliing any one of the sections at random. Point of contact is αφή, chord of contact ή τας άφας Ιττιζίν-^ννονσα. The point of intersection of two tangents is ή συμπτωσις των εφ- απτόμενων. The following elliptical expressions are found in Apollonius : απ" avTov ή ΔΒ Ιφαπτέσθω let ΔΒ be the tangent (draivn) from Δ (outside the curve) ; eav aV αύτοΰ ή μίν έφάπτηται, ή δέ τέμνη if {there he d/rawKi) from it (two straight lines of which) one touches, and the other cuts (the curve). 11. Asjnuptotes. Though the technical term used by Apollonius for the asymp- totes is ασύμπτωτος, it is to be observed that the Greek word has a wider meaning and was used of any lines which do not meet, in whatever direction they are produced. Thus Proclus*, quoting from Geminus, distinguishes between (a) ασύμπτωτοι which are in one plane and (b) those which are not. He adds that of ασύμπτωτοι which are in one plane " some are always at the same distance from one another (i.e. parallel), while others continually diminish the distance, as a hyperbola apj)roaches the straiglit line and the * Comment, in End. i. p. 177. Clxiv APPENDIX TO INTRODUCTION. conchoid the straight line." The same use of ασύμπτωτος in its general sense is found even in Apollonius, who says (ii. 14) πασών των άσνμπτωτων rg τομ.^ lyyiov (Ισιν αί ΑΒ, ΑΓ, tJie lines ΑΒ, ΑΓ (the asymptotes proper) are nearer than any of the lines which do not meet the section. The original enunciation of ii. 14 [Prop. "36] is interesting: αί ασύμπτωτοι καΐ η το/χτ; el<; απ^φον έκβαλλόμ^ναί eyyioV Τ€ προσά-γονσιν εαυταΐ? και παντός του Βοθεντος διαστ^/Αατο? fts ίλαττον άφικνοννται Βιάστημα, the asymptotes and the section, if produced to infinity, ajyjyroach nearer to one another and come within a distance less than any given distance. One of the angles formed by the asymptotes is η πψύχονσα την υπΐρβολην the angle containing (or including) the hyperbola, and similarly we find the expression ctti /aiS? των α'συ/Ατττώτων τών πζρίίχουσων την το/Ατ^ν on one of the asymptotes containing the section. The space between the asymptotes and the curve is ό άφοριζόμίνος τόπος υπό τών ασύμπτωτων και n7S τομής. 12. Data and hypotheses. Given is hoOiU or δεδο/^λενος ; given in positio7i θεσα δεδο/χεντ;, given in magnitude τω μ(•γΐθ^ δεδο/χε'ντ; (of straight lines). For is or will be, given in position we frequently find ^ε'σει εστίν, εσται without δεδο- /Αε'νος, or even ^εσει alone, as in ^ε'σει αρα η AE. A more remarkable ellipse is that commonly found in such expressions as πάρα ^ε'σει την ΑΒ, 2)Ci')'cdlel to ΑΒ (given) in 2)Ositio7i, and προς ^ε'σει ttj AB, used of an angle made with AB (given) in positio7i. Of hypotheses υπο'κειται and the other parts of the same verb are used, either alone, as in νποκείσθω τά μ\ν άλλα τά αυτά let all the other suppositions be the same, των αυτών νποκαμένων with the same suppositions, or Avith substantives or adjectives following, e.g. κύκλος υπόκειται η ΔΚΕΛ -γραμμή the line ΔΚΕΛ is by hypothesis a circle, υπόκειται Ίση is by hypothesis equal, υπόκεινται συμπίπτονσαι they meet by hypothesis. In accordance with the Avell-knovvn (ireek idiom δπερ ουχ υπόκειται means which is contrary to the hypothesis. 13. Theorems and problems. In a theorem loliat is required to be proved is sometimes denoted by TO προτίθεν, and the requirement in a problem is to ε'πιταχ^ε'ν. Thus ει μίν ουν η ΑΒ α^ων ε'στι, γεγονός αν ειτ; το ίπιταχθίν ij then ΑΒ is an axis, that which loas required would have been done. To draiv NOTES ON ΪΗΚ TERMINOLOGY OF ORKEK (ίΕΟΜΕΤΙΙΪ. clxv 171 the manner required is ayayitv ώς πρόκειται. When the solution of a problem has been arrived at, e.g. when a required tangent has been drawn, the tangent is said ττοιύν το πρόβλημα. In the ίκθίσκ: or setting out of a theorem the re-stateiiiont of what it is required to prove is generally introduced by Apollonius as well as Euclid by the words λέγω, on ; and in one case ApoUonius abbreviates the re-statement by saying simply λε'γω, otl Ισται τά τη% προτάσεως I say that the property stated in the enunciation triU be true ; it is to be proved is Set/cTcov, it renuiins to he proved λοιποί/ άρα SeiKT€ov, let it be required to dra^v hiov Ιστω dyayav. The synthesis of a problem regularly begins with the words συν- Τ€^7;σ£ται hrj (το πρόβλημα) όντως. li. Constructions. These are nearly always expressed by the use of the perfect imperative passive (with which may be classified such perfect imperatives as γεγονετω from γι'νεσ^αι, συΐ'εστατω from συησταναι, and the imperative κβίσθω from κίΐμαή. The instances in ApoUonius where active forms of transitive verbs appear in constructions are rare ; but we find the following, idv ττοιησωμ^ν if tve make (one line in a certain ratio to another), ό/ΐΛοιω? γαρ τω ττροίίρημίνω α'γαγών την ΑΒ ίφαπτομίνην λέγω, οτι for in the same manner as before, after draiviny the tangent AB, / say that..., επιζεΰ^αντες την ΑΒ epou/xev having joined AB we shall prove ; Λvhile in άγαγόντες yap ΐτηφανονσαν την ΘΕ εφάπτεται αυττ; we have a somewhat violent anacoluthon, /or, having drawn the taiigent ΘΕ, this touches. Of the words used in constructions the following are the most common : to dratv αγειν, διάγειν and other compounds, to join iirtCevy• nJvai, to produce έκβάλλαν, ττροσΐκβάλλίΐν, to take or supply πορίζαν, to cut off άτΓολαμβάν^ιν, άποτε/ζ,ΐ'ειν, αφαιρεί»', to construct συΐ'ΐστασ^αι, κατασκευά^ειν, to describe γρα'φω and its compounds, to apply παρα- βάλλίΐν, to erect άνιστάναι, to divide Staipetv, to bisect Βιχοτομ^ΐΐ'. Typical expressions are the follo\ving : rrj ΰπο των ΗΘΕ γωΐ'ΐα ίση σννίστάτω ή νπό των ΒΑΓ let the angle ΒΑΓ be constructed eqna^ to the angle ΗΘΕ ; ό κίντρω τω Κ διαστ7;/Λατι δε τω ΚΓ κνκλος ypaφόμtvo<; tic circle described icith Κ as centre and at a. distance ΚΓ ; άνεστα'τω άπό τ^ς AB επίπεδοι/ ορθόν ττρος το νττοκίίμίνην ΙττίττίΒον let α plane be erected 071 ΑΒ αϊ right angles to the supposed pi a 7ie ; κ(ίσθω avrrj Ιση let (α?ί a7igle) be made equal to it, Ικκύσθω let (a line, circle etc.) be set out, άφηρήσθω απ αντον τμήμα let a segme7it he cut off from it, των αντων κατασκενασθά'των ivith the saj7ie constructioti. clxvi APPENDIX TO INTRODUCTION. No detailed enumeration of the various perfect imperatives is necessary ; but -^ί-^ονίτω for suppose it done deserves mention for its elegance. Let it he conceived is νοίίσθω : thus νούσθω κώνος, ου κορνφη το Ζ σημίΐον let α cone be conceived whose apex is the jyoiiit Ία. A curious word is κλαω, meaning literally to break off and generally used of two straight lines meeting and forming an angle, e.g. of two straight lines drawn from the foci of a central conic to one and the same point on the curve, άπο τώΐ' Ε, Δ σημείων κζκλά- σθωσαν ττροζ την -γραμμην αί ΕΖ, ΖΔ, (literally) /rom the points Ε, Δ let ΕΖ, ΖΔ be broken short off against the curve. Similarly, in a propo- sition of ApoUonius quoted by Eutocius from the Άναλυό /icvos τόπος, the straight lines drawn from the given jwints to meet on the circum- ference of the circle are at άττό των ZoBkvTiMv σημείων ctti την περιφε'ρειαν του κύκΧον κΧωμεναι (,νθεΐαι. 15. Operations (Addition, Subtraction etc.). The usual woi'd for being added is πρόσκειμαι : thus 8ίχα τΐτμηται η ΖΘ κατά. το Μ προσκειμίνην έχουσα την ΔΖ, or ΖΘ is bisected in Μ and has ΔΖ added. Of a magnitude having another added to it the participle of ττροσλαμβάνίΐν is used in the same way as λιποίν for having something subtracted. Thus το KP λιττον rj προσλαβόν το BO Ισον εστί τω ΜΠ means KP minus or phis BO is equal to ΜΠ. μετά (with gen.) is also used for plus, e.g. το νπο AEB μετά τον άττο ΖΕ is equivalent to AE . EB + ZE^ A curious expi'ession is συναμφότερος η ΑΔ, ΔΒ, or συναμφότερος η ΓΖΔ meaning t/ie sum of Α^, ΔΒ, or ofTZ, ΖΔ. Of adding or subtracting a common magnitude Kotvo's is used : thus Kotvov προσκείσθω or αφηρησθω is let the common [magnitude) be added, or taken away, the adjective Aoitto's being applied to the remainder in the latter case. To exceed is ύπερβάλλειν or υττερεχ^είν, the excess is often η υπέροχη, ην υπερέχει, κ.τ.λ., ΠΑ exceeds ΑΟ by ΟΠ is το ΠΑ του ΑΟ υπερέχει τω ΟΠ , to differ from is 8ιαφερειν with gen., to differ by is expressed by the dative, e.g. (a certain triangle) differs from ΓΔΘ by the triangle on Pi® as base similar to ΓΔΛ, Βιαφερει του ΓΔΘ τω α'πό της ΑΘ τρινώνω όμοίω τω ΓΔΛ ; (the area) by which the square on ΓΡ differs from, the square on A2, ω διαφέρει το από ΓΡ του α'πο Α2. For multiplications and divisions the geometrical equivalents are the methods of proportions and the application of areas ; but of numerical multiples or fractions of magnitudes the following are NOTES ON THE TERMINOLOGY OF GREEK GKOMETUY. clxvii typical instances : the half of AB, η ημίσεα τη<; AB ; the fourth part of the figure, το τέταρτον τον «ίδους ; fou7• times the rectangle AE . ΕΔ, TO Τίτράκις υπό AEA. IG. Proportions. Ratio is λόγο?, tvill be cut in the same ratio «is τον αντον λόγο»' τμηθησονται, the three jrroportionals αί Tpcis άναλογον ; όβύ<</ a mean proportional between ΕΘ, ΘΑ, μίσον λόγον ίχονσα, or μίσ-η dvaXoyov, των ΞΘΑ. The sides about the right angles (are) jyroportional irepl ορθας γωνία? αί 7Γλ€υραι οΐναλογον. I'he ratio of A to Β is 6 λόγος, oV ίχίί το A ττρόζ τό Β, or ό rot• A ττρό? TO Β λόγο? ; stippose the ratio of ΓΔ to Δ Β made the same as the ratio of ΓΗ to HB, τω ττ7? ΓΗ ττρό? ΗΒ λόγω ό αύτοζ πίττοιησθω ό 7^7? ΓΔ ττρό? ΔΒ ; Α has to Β α greater (or less) ratio than Γ has to Δ, to A προζ TO Β μείζονα (or ελάσσονα) λόγον €χ€ί ητΓ€ρ τό Γ πρό? τό Δ, or του, oV ίχ€ΐ τό Γ ττρό? τό Δ; the ratio of the square of the inner segment to the square of the renipAning segment, λόγο?, ov Ιγζΐ τό εντό? τ/η7/Αα irpo% τό λοιττόν 8ννάμ€ΐ. The following is the ordinary form of a proportion : as the square on A2 is to the rectangle under B2, 2Γ so is ΕΘ to ΕΠ, ω? τό aVo A2 ττροζ τό υ'ττό Β2Γ, οΰτω? η ΕΘ ττρό? ΕΠ. In a proportion the antece- dents are τα τ^γοΰρ,ενα, i.e. the leading terms, the consequents τά €ΤΓΟμενα ; as one of the antecedents is to one of the consequents so are all the antecedents (taken together) to all the consequents (taken together) ω? cv των ηyovμevωv ττρο? €v των ίττομΐνων, ούτω? ατταντα τα ■ηγονμΐνα ττρο? ατταντα τα €ΤΓομ€να. Α very neat and characteristic sentence is that which forms the enunciation of Euclid v. 19 : eav y cj? όλον ττρό? όλον, οΰτω? άφαφΐθΐν •πτροζ άφαφ£^€ν, και τό λοιπόν πρό? τό λοιπόν Ισται ο5? όλον πρό? όλον. If as α whole is to a whole so is (a part) taken awaij to (a part) taken away, the r&mainder also will be to the remainder as the lohole to the whole. Similarly in Apollonius we have e.g. eVei ου ν ώ? όλον «στί τό από ΑΕ πρό? όλον τό ΑΖ, οΰτω? άφαιρίθϊν το υπό ΑΔΒ πρό? άφαιρ^θιν τό ΔΗ, και λοιπόν έστι πρό? λοιπόν, ώ? όλον πρό? όλον, since then, as the whole the square on AE is to the whole the (parallelogram) AZ, so is (the part) taken away the rectangle under ΑΔ, ΔΒ to (the part) taken away the (parallelogram) ΔΗ, remainder is also to remainder as whole to whole. To be compounded of is συγκασ^αι, the ratio compounded of 6 σνγκΐίμΐνοζ (or συνημμένος, from συνάπτειν) λόγο? (Ικ τ( τον, δν (χα κ.τ.λ.), the ratio cornpounded (<f the ratios) of tlie sides ό σvyκ(ίμ€yo<; clxviii APPENDIX TO INTRODUCTION. λόγος €K των ττλενρών. συγκΐ^σθαι is moreover used not only of being a compounded ratio, but also of being eqnal to a ratio compounded of two others, exen Avhen none of the terms in the two latter ratios are the same as either term of the fii'st ratio. Another way of describing the ratio compounded of two others is to use μ€τά (with gen.) which here implies multiplication and not addition. Thus ό 1-175 A2 ττρό? 2Γ λόγος /χίτά τον τη<; Α2 προς 2Β is the ratio compounded of the ratio of k'% to 2Γ and that of A2 to 2B. Similarly κοινοί άφΐ)ρησθω ό της ΓΔ προς ΓΘ means let the common ratio of ΓΔ to ΓΘ be divided out (and not, as usual, subtracted), KOLvov άφαίρΐθίντος τούτου του λόγου dividing out by this common ratio. Taking the rectangle contained by ΘΕ, EZ as a middle term is τον νπο ΘΕΖ μΙσον λαμβανομένον, taking AH as a common altitude της AH κοινοί) νψονς ΧαμβανομΙνης. So that the corresponding terms are continuous ώστ€ τάς όμολόγονς συνεχείς eivat ; so that the segments adjoining the vertex are corre- sponding teryns ώστε ομόλογα cTvai τα προς ttj κορνψ-τ) τ/χ,ι/'/χ,ατα. There remain the technical terms for transforming such a pro- portion as a :b = c : d. These correspond with the definitions at the beginning of Eucl. Book v. Thus εναλλάξ alternately (usually called permutando or alternando) means transforming the proportion into a : c-b : d. άνάπαλίν reversely (usually invertendo), b : a==d : c. σννθίσίς λόγου is composition of a ratio, by which the ratio a : b becomes a + b : b. The corresponding Greek term to compo- nendo is συν^εντι Avhich means no doubt, literally, " to one who has compounded," or " if we compound," the ratios. Thus σννθΐντί is used of the inference that a + b :b = c + d: d. διαιρεσις λόγου means divisio7i of a ratio in the sense of separation or subtraction in the same way as συν^ίσις signifies addition. Similarly διελόντι (the translation of which as divide^ido or dirimsndo is misleading) means really separating in the sense of subtracting : thus a - b : b = c — d : d. ανάστροφη λόγου conversion of a ratio and άναστρίφαντί conver- tendo correspond respectively to the ratio a la-b and to the inference that a : a — b = c : c — d. St ίσον, generally translated ex aequali (sc. distantia), is applied to tlie inference e.g. from the proportions a : b : c : d etc. = ^1 : Β : C : D etc. that a : d= A : D. All tlie expressions above explained, Ιναλλάξ, (Ινύτταλιΐ', σννθίντι, NOTES ON THE TERMINOLOGY OF GREEK GEOMETRY. clxix δΐίλόντί, άναστρίψαντι, 8l Ισον are constantly used in Apollonius as in Euclid. In one place we find the variant 8ia 8e το ανάτταλι»'. Are in recijrrocal pi-ojiortion is α.ντηΓ€πόνθασιν. 17. Inferences. The usual equivalent for therefore is αρα, e.g. iv rrj €Vi0avcta άρα Ιστί it is therefore on the surface, iidda apa iariv τ; A Β therefore AB is a straight line ; ovv is generally used in a somewhat weaker sense, and in conjunction with some other word, in order to mark the starting point of an ai'gument rather than to express a formal inference, so that Λνβ can usually translate it by then, e.g. «Vei oty since, then, on μ\ν ονν...φαν(ρόν it is, then, clear that.... 8η is some- what similarly used in taking up an argument. .SO that is ώστ£, that is τουτέστιν. A corollary is often introduced by και φαΐίροΊ•, oTt, or by συναττοδίδεικται it is proved at the same time. It is at once clear φανερον αυτο'^εν, from this it is clear ΐκ δτ/ τούτου φαι /cpo'v, for this reason δια τούτο, for the same reason δια τά αύτύ, wherefore διόπίρ, in the same way as above or before κατά τά αυτά TOts επάνω or (.ρ.προσΘΐν, similarly it will be shown ομοίως και δ€ΐ;!(^77σ€ται, the same results as before will follorv τά αυτά τοις πρότΐρον σνμβησΐται, the same proofs tvill apply αί αΰται άττοδει^εις άρμόσονσι. Conversely αντιστρόφως, by the converse of the theorem διά τ-ην αντιστροφην του θίωρηματος, by what zvas j^'oved and its converse διά τά elp -ημίνα και τά αντίστροφα αυτών. By w/tat was before proved in the case of the hyperbola 8ia. το προ8(.8ίΐ•γμ(.νον cttI rijs ΰττίρβοληζ ; for the same (facts) have been proved in the case of the jmrallelograms which are t/ieir doubles και γαρ CTTi των διττλασιων αυτών τταραλληλογράμμων τά αυτά δεδεικται. By the similarity of the triangles διά τ^ν ομοιότητα των τρίγωνων, by parallels διά τά? τταραλληλονς, by the {])ropei-ty of the) section, parabola, hyjyerbola διά τ^ν τομην, τταραβολην, νπΐρβολην. The properties ivhich have already been proved true of t/ie sections when the original diameters are taken (as axes of reference) όσα ττροδεδεικται ττερι τά? τομας συμβαίνοντα σνμΊΓαραβαλλομ€ΐων των αρχικών 8ιαμ€τρων. Much more ττολλώ μάλλον. Cf. ττολύ πρότίρον τίμνίΐ την τομην much sooner does it cut the section. 18. Conclusions. Which it teas rpquired to do, to prove οττερ 18ίΐ ποιησαι, δεΐ^αι ; which is absurd όπερ άτοττον ; and this is impossible, so that the H. C. in clxx APPENDIX TO INTRODUCTION. original supposition is so also τοντο δέ αδννατον ώστ€ καΐ το Ιξ αρχηζ. A7id again the absurdity tvill he similarly inferred και ττάλιν ομοίως σνναχθησ€ται το άτοπον. 19. Distinctions of cases. Tliese properties are general, hut for the hyperbola only etc. ταΰτα μ\ν κοινώς, Ιτη δέ της νττΐρβολης μόνης κ.τ.λ., in the third figure «πι τ-^ς τρίτης καταγραφής or τον τρίτον σχήματος, in all the possible cases κατά πάσας τάς (ν^^χομ^νας διαστολάς. 20. Direction, concavity, convexity. In both directions έφ' Ικάτερα, totvards the same parts as the section «πι ταΰτά τ^ τομτ) ; towards the direction of the point E, inl τα μέρη, ίφ* α £στι το Ε ; οη the same side of the centre as AB, hrl τά αυτά μ^ρη τον κίντρου, iv οΐς c<mv η ΑΒ. There is also the expression κατά τά Ιπόμΐνα μέρη της τομής, meaning literally in the succeeding 2)arts of tJie section, and used of a line cutting a branch of a hyperbola and passing inside. The concave parts τά κοίλα, the convexities τά κνρτά, not having its concavity (convexity) toioards the same parts μη ΙπΙ τά αυτά μίρη τά κοίλα (τά κυρτά) «χουσα, towards the same ])arts as the concavity of the curve €7ri τά αυτά τοΙς κοίΧοις της -γραμμής, if it touc/oes with its concave side iav ίφάπτηται τοις κοίλοις αντης, will touch on its concave side Ιφάφίται κατά τά κοίλα. Having its convexity turned the opposite loay άνίστραμμίνα τά κυρτά ί^ουσα. 21. Infinite, Infinity. Unlimited or infinite άπειρος, to increase without limit or indefi- nitely (Ις άπειρον αΰ^άν€σ^αι. απίΐρος is also used in a numerical sense ; thus in the same way we shall find an infinite number of diameters τω hi αΰτώ τρο'ττω και άπίίρονς (.νρησομΐν Βιαμίτρονς. THE CONICS OF AP0LL0NIU8. ' TFTHF i UKI Vrr, 3ITY, THE CONE. If a straight line indefinite in length, and passing always through a fixed point, be made to move round the circumference of a circle Avhich is not in the same plane with the point, so as to pass successively through every point of that circumference, the moving straight line will trace out the surface of a double cone, or two similar cones lying in opposite directions and meeting in the fixed point, which is the apex of each cone. The circle about which the straight line moves is called the base of the cone lying between the said circle and the fixed point, and the axis is defined as the straight line drawn from the fixed point or the apex to the centre of the circle forming the base. The cone so described is a scalene or oblique cone except in the particular case where the axis is perpendicular to the base. In this latter ca,se the cone is a right cone. If a cone be cut by a plane passing through the apex, the resulting section is a triangle, two sides being straight lines lying on the surface of the cone and the third side being the straight line which is the intersection of the cutting plane and the plane of the base. Let there be a cone Avhose apex is A and whose base is the circle BC, and let be the centre of the circle, so that .4 is the axis of the cone. Suppose now that the cone is cut by any plane parallel to the plane of the base BC, as DE, and let H. c. I THE roxirs OF APOLLONIUS. the axis Λ meet the plane DE in o. Let ρ be any point on the intersection of the plane DE and the surface of the cone. Join Ap and produce it to meet the circumference of the circle BC in P. Join OP, op. Then, since the plane passing through the straight lines Λ0, ΛΡ cuts the two parallel planes BG, DE in the straight lines OP, op respectively, OP, op are parallel. .•. op: OP = Ao:AO. And, BPG being a circle, OP remains constant for all positions of ^j on the curve DpE, and the ratio Ao: A is also constant. Therefore op is constant for all points on the section of the surface by the plane DE. In other words, that section is a circle. Hence all sections of the cone ivhich are parallel to the circular base are circles. [I. 4.] * Next, let the cone be cut by a plane passing through the axis and perpendicular to the plane of the base BG, and let the section be the triangle ABG. Conceive another plane Η Κ drawn at light angles to the plane of the triangle ABG and cutting off from it the triangle AHK such that AHK is similar to the triangle ABG but lies in the contrary sense, i.e. such that the angle ΑΚΗ is equal to the angle ABG. Then the section of the cone by the plane HK is called a subcontrary section (νττβναντία τομή). * The references in this form, here and throughout the book, arc to the original propositions of ApoUonius. THE CONE. 3 Let Ρ be any point on the intersection of the plane II Κ with the surfiice, and F any point on the circumference of the circle BG. Draw PM, FL each perpendicular to the plane of the triangle ABC, meeting the straight lines HK, BG respec- tively in M, L. Then PM, FL are parallel. Draw through Μ the straight line BE parallel to BG, and it follows that the plane through DME, PM is parallel to the base BG of the cone. Thus the section DPE is a circle, and DM. ME= PM\ But, since DE is parallel to BG, the angle AD Ε is equal to the angle ABG which is by hypothesis equal to the angle ΑΚΗ. Therefore in the triangles iri)if, EKM the angles HDM, EKM are equal, as also are the vertical angles at M. Therefore the triangles HDM, EKM are similar. Hence HM : MD = EM : MK. .•. HM.MK = DM.ME = PM\ And Ρ is any point on the intersection of the plane HK with the surface. Therefore the section made by the plane HK is a circle. Thus they-e are two senes of circular sections of an oblique cone, one series being parallel to the base, and the other consisting of the sections subcontrary to the first series. [I. 5.] Suppose a cone to be cut by any plane through the axis making the triangular section ABG, so that BG is a diameter of the circular base. Let Η be any point on the circumference of the base, let HK be perpendicular to the diameter BG, and let a parallel to Η Κ be drawn from any point Q on the surface of the cone but not lying in the plane of the axial triangle. Further, let AQha joined and produced, if necessary, to meet 1—2 4 THE COXICS OF APOLLONIUS. the circumference of the base in F, and let FLF' be the chord perpendicuhir to BG. Join AL, AF'. Then the straight line through Q parallel to HK is also parallel to FLF' ; it follows therefore that the parallel through Q will meet both AL and AF'. And AL is in the plane of the axial triangle ABC. Therefore the parallel through Q will meet both the plane of the axial triangle and the other side of the surface of the cone, since AF' lies on the cone. Let the points of intersection be V, Q' respectively. Then QV:VQ' = FL: LF', and FL = LF'. .•. QV= VQ', or QQ' is bisected by the plane of the axial triangle. [I. C] Again, let the cone be cut by another plane not passing through the apex but intersecting the plane of the base in a straight line DME perpendicular to BC, the base of any axial triangle, and let the resulting section of the surface of the cone be DPE, the point Ρ lying on either of the sides AB, AG o( the axial triangle. The plane of the section will then cut the plane of the axial triangle in the straight line PiU joining Ρ to the middle point of DE. Now let Q be any point on the curve of section, and through Q draw a straight line parallel to DE. Then this parallel will, if produced to meet the other side of the surface in Q', meet, and be bisected by, the axial THE CONE. triangle. But it lies also in the plane of the section DPE\ it will therefore meet, and be bisected by, PM. Therefore PM bisects any chord of the section which is parallel to DE. Now a straight line bisecting each of a series of parallel chords of a section of a cone is called a diameter. Hence, if a cone he cut by a plane which intersects the circuku' base in a straight line })erpendicular to the base of any axial triangle, the intersection of the cutting plane and the plane of the axial triangle will be a diameter of the resulting section of the cone. [I. 7.] If the cone be a right cone it is clear that the diameter so found will, for all sections, be at right angles to the chords which it bisects. If the cone be oblique, the angle betAveen the diameter so found and the parallel chords which it bisects will in general not be a right angle, but will be a right angle in the particular case only where the plane of the axial triangle ABC is at right angles to the plane of the base. Again, if PJ/ be the diameter of a section made by a plane cutting the circular base in the straight line DME perpen- dicular to BC, and if PJ/ be in such a direction that it does not meet AC though produced to infinity, i.e. if Ρ Μ be either parallel to AC, or makes with PB an angle less than the angle Β AC and therefore meets CA produced beyond the apex of the cone, the section made by the said plane extends to infinity• 6 THE COXICS OF APOLLONIUS. For, if we take any point V on PM produced and draw through it HK parallel to BC. and QQ' parallel to DE, the plane through HK, QQ' is parallel to that through DE, BC, i.e. to the base. Therefore the section HQKQ' is a circle. And D,E,Q,Q' are all on the surface of the cone and are also on the cutting plane. Therefore the section DPE extends to the circle HQK, and in like manner to the circular section through any point on PM produced, and therefore to any distance from P. [I. 8.] [It is also clear that ΒλΡ = BM.MC, and QV" = HV. VK : and HV . VK becomes greater as V is taken more distant from P. For, in the case where Ρ Μ is parallel to AC, VK remains constant while HV increases ; and in the case where the diameter PM meets CA produced beyond the apex of the cone, both HV, VK increase together as V moves aAvay from P. Thus QV increases indefinitely as the section extends to infinity.] If on the other hand Ρ Μ meets AC, the section does not extend to infinity. In that case the section will be a circle if its plane is parallel to the base or subcontrary. But, if the section is neither parallel to the base nor subcontrary, it Λνΐΐΐ not be a circle. [I. 9.] For let the plane of the section meet the plane of the base in DME, a straight line perpendicular to BC, a diameter of the THE CONE. 7 circular base. Take the axial triangle through BC meeting the plane of section in the straight line PP'. Then P, P\ Μ are all points in the plane of the axial triangle and in the plane of section. Therefore PP' Μ is a straight line. If possible, let the section PP' be a circle. Take any \unn\, Q on it and draw QQ' parallel to DME. Then if Qi^' meets the axial triangle in V, QV= VQ'. Therefore PP' is the diameter of the supposed circle. Let HQKQ' be the circular section through Q(^' parallel to the base. Then, from the circles, QV = HV. VK, QV' = PV.VP'. .•.HV.VK = PV.VP', so that HV: VP = P'V: VK. .•. the triangles VPH, VKP' are similar, and /.PHV = ^KP'V; .. ZKP'V = ZABC, and the section PP' is subcontrary : which contradicts the hypothesis. .•. PQP' is not a circle. It remains to investigate the character of the sections mentioned on the preceding page, viz. (a) those which extend to infinity, (b) those Avhich are finite but are not circles. Suppose, as usual, that the plane of section cuts the circular base in a straight line D.VE and that ABC is the axial triangle 8 THE COXICS OF APOLLONIUS. Avhose base BG is that diameter of the base of the cone which bisects DME at right angles at the point ^f. Then, if the plane of the section and the plane of the axial triangle intersect in the straight line PM, PM is a diameter of the section bisecting all chords of the section, as QQ', which are drawn parallel to BE. If QQ is so bisected in V,QV is said to be an ordinate, or a straight line drawn ordinate-'wise (τβτα'γμενως κατη'γμάνη), to the diameter PAi ; and the length PV cut olf from the diameter by any ordinate Q V will be called the abscissa of Q V. Proposition 1. [I. 11.] First let the diameter Ρ Μ of the section he parallel to one of the sides of the axial triangle as AC, and let QV be any ordinate to the diameter PM. Then, if a straight line PL (supposed to be draiun p)erpendicidar to PM in the plane of the section) be taken of such a length that PL : PA = BC'^ : Β A .AC, it is to be proved that QV' = PL.PV. Let HK be draAvn through V parallel to BC. Then, since QF is also parallel to DE, it follows that the plane through H, Q, Κ is parallel to the base of the cone and therefore THE CONE. 9 produces a circular section whose diameter is UK. Also QV is at right angles to HK. .•. HV.VK = QV\ Now, by similar triangles and by parallels, HV:PV=BC:AC and VK:PA=BC:BA. .•. HV. VK.PV.PA=BG':BA.AG. Hence QV .PV .PA = PL : PA = PL.PV:PV.PA. .•. QV'' = PL.PV. It follows that the square on any ordinate to the fixed diameter PM is equal to a rectangle applied (τταραβάλΧβιν) to the fixed straight line PL drawn at right angles to PM with altitude equal to the corresponding abscissa PV. Hence the section is called a Parabola. The fixed straight line PL is called the latus rectum (ορθία) or the parameter of the ordinates (παρ' ην δύ- νανται αϊ Karar^opLevaL τ€τα'^μίνω<;). This parameter, corresponding to the diameter PM, will for the future be denoted by the symbol ;λ Thus QV' = p.PV, or QV'^cPV. Proposition 2. [I. 12.] Next let Ρ Μ not be parallel to AC but let it meet CA produced beyond the apex of tJie cone in P'. Draw PL at Hght angles to Ρ Μ in the plane of the section and of such a length that PL : PF = BF . FG : AF\ where AF is a straight line through A parallel to Ρ Μ and meeting BG in F. Tlien, if VR be drawn parallel to PL and P'L be joined and produced to meet VR in R, it is to be proved that QV' = PV.VR. As before, let HK be drawn through V parallel to BG, so that QV' = HV.VK. 10 THE COXICS OF APOLLONIUS. Then, by similar triangles, HV:PV=BF:AF, VK :P'V=FC:AF. .•. HV.VK :PV.P'V= BF.F('.AF\ Hence QV :PV .P'V=PL .PP' = VE:P'V = PV.VR:PV.P'V. .•. QV' = PV.VR. It follows that the square on the ordinate is equal to a rectangle whose height is equal to the abscissa and Avhose base lies along the fixed straight line PL but overlaps (νττβρβάΧΧβι) it by a length equal to the difference between VR and PL*. Hence the section is called a Hyperbola. * Apollonius describes the rectangle PR as applied to the latus rectum but exceeding by a figure similar and similarly situated to that contained by I'l^ and PL, i.e. exceeding the rectangle VL by the rectangle LR. Thus, if QV=y, Py=x, PL=p, and PP':^d, y-=px + ^.x-, which is simply the Cartesian equation of the hyperbola referred to oblique axes coneiatiug of a diameter and the tangent at its extremity. THE CONE. 11 PL is called the latus rectum or the parameter of the ordinates as before, and PP' is oallcil the transverse ( /; TrXayia). The fuller expression transverse diameter ( /; -rrXayia δίά/ΐ6τρο9) is also used; and, even more commonly, Apullunius speaks of the diameter and the corresponding parameter together, calling the latter the latus rectum (i.e. the erect side, η ορθία ifkevpa), and the former the transverse side {η irXayia TrXeupa), of the figure (β'δος) on, or applied to, the diameter {ιτρος rrj Βιαμέτρω), i.e. of the rectangle contained by PL, PP' as drawn. The parameter PL will in future be denoted by jj. [Coil. It follows from the proportion QV':PV.P'V=PL:PP' that, for any fixed diameter PP', QV iPV.P'Visa constant ratio, or QF•^ varies as PF.P'F.] Proposition 3. [I. 13.] If Ρ Μ meets AC in P' and BG in M, draw A F parallel to PM nieetiiuj BG produced in F, and draw PL at right angles to PM in the plane of the section and of such a length that PL : PP' = BF.FC : AF\ Join P'L and draw VR parallel to PL meeting P'L in R. It luill he proved that QV"' = PV.VE. y 12 THE COXICS OF APOLLOXIUS. Draw HK through V parallel to BC. Then, as before, QV' = HV. VK. Now, by similar triangles, HV.PV=BF:AF, VK:P'V = FG:AF. .•. HV.VK:PV.P'V = BF.FC :AF\ Hence QV : PV . P'V= PL : PP' = VR:P'V = PV. VR.PV.P'V. .•. QV' = PV.VE. Thus the aquarc on the ordinate is equal to a rectangle whose height is equal to the abscissa and Avhose base lies along the fixed straight line PL but falls short of it (iWeiTrei) by a length equal to the difference between VR and PL*. The section is therefore called an Ellipse. As before, PL is called the latus rectum, or the para- meter of the ordinates to the diameter PP', and PP' itself is called the transverse (with or without the addition of diameter or side of the figure, as explained in the last proposition). PL will henceforth be denoted by p. [Cor. It follows from the proportion QV':PV.PV' = PL:PP' that, for any fixed diameter PP', QV^:PV.P'V is a constant ratio, or QV varies SisPV.PV.] * Apollonius describes the rectangle PR as applied to the latiu rectum but falling short by a figure similar and similarly situated to that contained by PP" and PL, i.e. falling short of the rectangle VL by the rectangle lAi. If QV=y, PV=x, PL=p, and PP' = d, y-=px Thus ApoUouius' enunciation simply expresses the Cartesian equation referred to a diameter and the tangent at its extremity as (oblique) axes. THE CONE. 18 Proposition 4. [I. U.] If a plane cuts both parts of a double cone and does not pass through the apex, the sections of the two parts of the cone will both be hyperbolas which will have the same diameter and equal later-a recta coiTesponding thereto. And such sections are called OPPOSITE BRANCHES. Let BChe the circle about which the straight line generating the cone revolves, and let B'C be any parallel section cutting the opposite half of the cone. Let a plane cut both halves of the cone, intersecting the base BC in the straight line DE and the plane B'C in D'E\ Then ΌΈ' must be parallel to DE. Let BC be that diameter of the base which bisects DE at right angles, and let a plane pass through BC and the apex A cutting the circle B'C in B'C, which will therefore be a diameter of that circle and will cut D'E' at right angles, since B'C is parallel to BC, and DE' to DE. ^* THE COXICS OF APOLLONIUS. Let ^.4i?^' be drawn through A parallel to MM', the straight Hr^ join.ng the n.ddle points of DE, D'E' and meeting C^ iiA respectively in P, P'. ^ ' Draw perpendiculars PL, P'L to MM' in the plane of the section and of such length that PZ ■.PP' = BF.FG:AF\ P'L':P'P=B'F'.F'C':AF'\ Since now ifP, the diameter of the section DPE when Sl^aT/plh'ofa/^'" ^^^-^ ^^^ ^^-' ^^^ -'^" Also since i)'^' is bisected at right angles by the base of he axial triangle AB'C, and M' p\n the^lane of he lia triangle meets C'A produced beyond the anex A fh! Τ DPE' i, also a hyperbola. ^ ^ ^' '^" ■''^'^"" And the two hyperbolas have the same diameter MPP'M. It remains to prove that PL = P'L'. We have, by similar triangles, BF:AF=B'F':AF', FC :AF=F'C' :AF'. •■BF.FC.AF' = B'F' . F'C : AF'\ Hence pi . pp' ^ p.^, . p,p -.PL^P^L'. THE DIAMETER AND ITS CONJUGATE. Proposition 5. [I. 15.] If through C, the middle point of the diameter PP' of (oi ellipse, a double ordinate BCD' he draiun to PP', BCD' will bisect all chords parallel to PP', and will tJierefore he a diameter the ordinates to which are parallel to PP'. In other words, if the diameter bisect all chords parallel to a second diameter, the second diameter will bisect all chords parallel to the first. Also the parameter of the ordinates to BOB' will he a third proportional to BB', PP'. (1) Let QF be any ordinate to PP' , and through Q draw QQ parallel to PP' meeting BB' in ν and the ellipse in Q' \ and let Q V be the ordinate drawn from Q to PP'. 16 THE comes OF APOLLONIUS. I Then, if PL is the parameter of the ordinates, and if ΡΈ is joined and VR, CE, V'R' draAvn parallel to PL to meet P'L, we have [Prop. 3] QV' = PV. VR, Q'V'^PV'.V'R'; and QV = QV, because QV is parallel to Q'V and QQ' to PP'. .■.PV.VR = PV'.V'R. Hence PV : PV'=V'R': VR = P'V : P'V. .'. PV: PV''-PV=P'V' : P'V- P'V, or PV:VV' = FV':VV'. ..PV=P'V. Also GP=CP'. By subtraction, CV = CV\ and .•. Qv = vQ', «o that QQ' is bisected by BD'. (2) Draw i^A" at right angles to DD' and of such a length that DB' : PP' = PP' : DK. Join D'A and draw vi' parallel to DK to meet D'A^ in r. Also draw Ti?, Xi^if and ES parallel to PP'. Then, since PC =CP', PS = SL and CE=EH; .•. the parallelogram (P^) = (>Sri/). . Also (PP) = ( VS) + (8R) = (SU) + (RH). By subtraction, (PA) - (PR) = (PA) ; .■.GO'-QV' = RT.TE. But CP- - Q F•' = CP•' - Ov' = P'y . vD. .■.D'v.vD = RT.TE (A). Now PP' : PP' = PP' : PA, by hypothesis. .■.DD' :DK = DD":PP"' = CD' : GP' = PG.GE:GP' = RT. TE : RT\ and DD' : P7i^ = D'v : vr ν = D'v .vD : vD. vr ; ,' .•. D'v .vD:Dv.vr = RT.TE: RT\ i But D'v .vD = RT. TE, from (A) above ; ^^ .•. Dv.vr = RT = CV'=Qv\ I THE DIAMETER AND ITS CONJUGATE. 17 Thus DK is the parameter of the ordinates to DD', such as Qv. Therefore the parameter of the ordiuates to DD' is a third proportional to DD', PF. Cor. We have 00"" = PG.GE = hPP'.\PL; ..DD" = PP'.PL, or PP' : DD' = DD' : PL, and PL is a third proportional to PP', DD'. Thus the relations of PP', DD' and the corresponding parameters are reciprocal. Def. Diameters such as PF, DD', each of which bisects all chords parallel to the other, are called conjugate diameters. Proposition 6. [I. 16.] If from the middle jjoiiit of the diameter of a hyperbola with two branches a line be drawn parallel to the ordinates to that diametei-, the line so draimi ivill be a diameter conjugate to the former one. If any straight line be drawn parallel to PP', the given diameter, and meeting the two branches of the hyperbola in Q, Q' respectively, and if from C, the middle point of PP', a straight line be drawn parallel to the ordinates to PF meeting QQ' in V, we have to prove that QQ' is bisected in v. Let QV, Q'V be ordinates to PF, and let PL, FL be the parameters of the ordinates in each bmnch so that [Prop. 4] H. c. 2 18 THE CONICS OF ArOLLONIUS. PL = FL'. Draw VR, V'R parallel to PL, P'L', and let PL, P'L be joined and produced to meet V'R, VR respectively in R',R. Then we have QV'^PV.VR, qV" = PV' .V'R. .'. PV. VR = P'V . V'R, and V'R :VR = PV:P'V'. Also PV : V'R = PR : RL' = RP : PL = P'V : VR. .•. PV ■.P'V=V'R' .VR = PV. RV, from above ; ... PV '.PV=P'V:P'V', and PV + PV : PV = RV + RV : RV, or VV ■.PV=VV':RV'; Λ PF=P'F'. But CP = CR; .•. by addition, CF=CF', or Qv = Q'v. Hence Gv is a diameter conjugate to PR. [More shortly, we have, from the proof of Prop. 2, QV:PV.P'V=PL:PP', Q'V" :RV.PV = P'L' : PR, and QV=Q'V, PL = P'L': .•. PV.RV=PV.RV', or PF : PF' = P'F' : P'F, whence, as above, PV= P'V'.] Def. The middle point of the diameter of an ellipse or hyperbola is called the centre; and the straight line dmwn parallel to the ordinates of the diameter, of a length equal to the mean proportional between the diameter and the parameter, and bisected at the centre, is called the secondary diameter {8evTepa Βιάμβτρος). Proposition 7. [I. 20.] In a parabola the square on an ordinate to the diamete?' vanes as the abscissa. This is at once evident from Prop. 1. THE DIAMKTFU AND ITS CONMIYIATE. 19 Proposition 8. [I. -21.] In a hi/perhohi, an ellipse, or <i circle, if QV be ani/ nrdindte to the diameter PP', QV'xPV.P'V. [This property is at once evident from the proportion QV':PV.P'V=PL:PP' obtained in the course of Props. 2 and 3 ; but ApoUonius gives a separate proof, starting from the property QV^ = PV .VR which forms the basis of the definition of the conic, as follows.] Let QV, Q'V be two oi-dinates to the diameter PP'. Then QV' = PV.VR, qV^PV. V'R'; .•. QV ■.PV.PV= PV.VR : PV.P'V = VR :P'V=PL:PP'. 2—2 20 THE COXICS OF APOLLONIUS. Similarly QT* : PV'.FV = PL : PP'. .•. QV':Q'V"' = PV.P'V:PV'.P'V'; and QV^ : PV .P'V is Ά constant ratio, or QV'ocPV.P'V. Proposition 9. [I. 29.] If a straight line through the centre of a hi/perbola with two branches meet one branch, it will, if produced, meet the other also. Let PP' be the given diameter and C the centre. Let CQ meet one branch in Q. Draw the ordinate QV to PP', and set off GV along PP' on the other side of the centre equal to CV. Let V'K be the ordinate to PP' through V. We shall prove that QGK is a straight line. Since CF= CV, and CP = GP', it follows that PV= P'V ; .•. PV.P'V = PT.PV'. But QV : KV" = PV.P'V:FV'. PV. [Prop. 8] .•. QV=KV'; and QV, KV are parallel, while GV = GV. Therefore QGK is a straight line. Hence QG, if produced, will cut the opposite branch. THE DIAMETER AND ITS CONJUGATE. 21 Proposition lO. [I. 30.] any chord through the centre In a hyperbola or an is bisected at the centre. Let PP' be the diameter and G the centre ; and let QQ' be any chord through the centre. Draw the ordinates QV, Q'V to the diameter PP'. Then PV. P'V: P'V. PV = QV : Q'V' = (77^ : GV'\ by similar triangles. .•. CV'±PV.P'V•. CV' = CV"±P'V\PV' : GV (where the upper sign applies to the ellipse and the lower to the hyperbola). .•. GP' : GV = GP" : GV'\ But GP' = GP"; .•. CV'=GV'', and GV = GV'. And QV, Q'V are parallel ; .•. GQ=CQ:. TANGENTS. Proposition 11. [I. 17, 32.] If a straight line he draxmi through the extremity of the diameter of any conic parallel to the ordinates to that diameter, the straight line will touch the conic, and no othei' straight line can fall hetiueen it and the conic. It is first proved that the straight line drawn in the manner described will fall without the conic. For, if not, let it fall within it, as PK, where PM is the given diameter. Then KP, being drawn from a point Κ on the conic parallel to the ordinates to PM, will meet PM and will be bisected by it. But KP produced falls without the conic ; therefore it Avill not be bisected at P. Therefore the straight line PK must fall without the conic and will therefore touch it. It remains to be proved that no straight line can fall between the straight line drawn as described and the conic. (1) Let the conic be a parabola, and let PF be parallel to the ordinates to the diameter PV. If possible, let PK fall between PF and the parabola, and draw KV parallel to the ordinates, meeting the curve in Q. Then KV':PV''>QV' : PV >PL.PV:PV' >PL:PV. Let V be taken on Ρ Υ such that KV:PV' = PL.PV', and let V'Q'M be drawn parallel to QV, meeting the curve in Q' and PK in .1/. TANfiKNlS. Then KV'.PV'^FL-.PV = PL.rV': PV" = q'V"\PV'\ and KV • PV = MV" : PV'\ by parallels. Therefore MV" = Q'V'\ and MV = Q'V. Thus PK cuts the curve in Q', and therefore does not fall outside it : which is contrary to the hyi^othesis. Therefore no straight line can fall between PF and the curve. (2) Let the curve be a hyperbola or an ellipse or a cifxle. κ Let PF be parallel to the ordinates to PP', and, if pussible, let PK fall between PF And the curve. Draw KV parallel to the ordinates, meeting the curve in Q, and draw VR per- 24 THE COXICS OF APOLLONIUS. pendicular to PV. Join P'L and let it (produced if necessary) meet VR in R. Then QV = PV. VR, so that KV > PV. VR. Take a point S on VR produced such that KV' = PV.VS. Join PS and let it meet P'R in R'. Draw R'V parallel to PZ meeting PF in V, and through V draw VQ'ili parallel to QV, meeting the curve in Q' and PK in i¥. Now so that KV' = PV.VS, .•. VS:KV=KV:PV, VS:PV=KV':PV\ Hence, by parallels, VR' :PV' = iyV":PV", or Μ V is a mean proportional between Ρ V, VR', i.e. MV" = PV'.V'R' = Q' V, by the property of the conic. .•. MV' = Q'V'. Thus PK cuts the curve in Q', and therefore does not fall outside it : which is contrary to the hypothesis. Hence no straight line can fall between PF and the curve. TANGENTS. 2δ Proposition 12. [Ι. 33, 35.] If a point Τ be taken on the diameter of a parabola outside the curve and such that TF = PV, where V is the foot of the ordinate from Q to the diameter FV, the line TQ will touch the parabola. We have to prove that the straight line TQ or TQ produced does not fall within the curve on either side of Q. For, if possible, let K, a point on TQ or TQ produced, fall within the curve*, and through Κ draw Q'KV parallel to an ordinate and meeting the diameter in V and the curve in q. Then Q'F'^QF^ >KV'^: QV\ by hypothesis. > TV'"- : TV\ .-.PV .PV>TV'"- : TV\ Hence 4>TP .PV : VTP . PV > TV" : TV\ and, since TP = PV, ^TP.PV=TV\ .'.^TP.PV'>TV'\ But, since by hypothesis TF' is not bisected in P, ^TP.PV <TV'\ (which is absurd. Therefore TQ does not at any point fall within the curve, and is therefore a tangent. * Though the proofs of this pioposition and tlie uext follow //; form the method of reductio ad absurdtim, it is easily seen that they give in fact the direct demonstration that, if A' is any point on the tangent other than Q, the point of contact, A' lies outside the curve hecause, if KQ'V' be parallel to QV, it is proved that KV" >Q'V'. The figures in both propositions have accordingly been drawn in accordance with the facts instead of representing the incorrect assumption which leads to the iibsurdity in each liise. 2ϋ THE COyias OF APOLLUNIUS. Conversely, if the tangent at Q meet the diameter jif'oduced .outside the curve in the point T, Τ Ρ = PV. Also no straight line can fall bettveen TQ and the curve. [ApoUonius gives a separate proof of this, using the method of reductio ad absurdum.] Proposition 13. [I. 34, 36.] In a hyperbola, an ellipse, or a circle, if PP' be the diameter and QV an ordinate to it from a point Q, and if a point Τ be taken on the diameter but outside the curve such that TP : TP' = PV : VP', then the straight line TQ will touch the cm^e. We have to prove that no point on TQ or TQ produced falls within the curve. TANGENTS. 27 If possible, let a point Κ on TQ or T(^ produced fall within the curve*; draw Q'KV parallel to an ordinate meeting the curve in Q'. Join P'Q, V'Q, producing them if necessary, and draw through P' , Ρ parallels to TQ meeting V'Q, VQ in /, and H, Ν respectively. Also let the parallel through Ρ meet P'Q in M. Now, by hypothesis, ΡΎ : PV= TP' : TP ; .•. by parallels, P'H : PN = P'Q : QM = P'H:NM. Therefore PN = NM. Hence Ρ Ν . Ν Μ > ΡΟ . 0.1/, or ΝΜ:ΜΟ>ΟΡ:ΡΝ; .: ΡΉ : ΡΊ > OP : ΡΝ, or ΡΉ.ΡΝ>ΡΊ.ΟΡ. It follows that Ρ' Η. ΡΝ : 'PQ' > ΡΊ .OP : 'fQ'\ .•. by similar triangles P'V . PV : ΊΎ' > P'V .PV : ΊΎ", or P'V.PV:P'V'.PV'>TV':TV"; .'.QV':Q'V">TV':TV" >QV':KV'\ .•. Q'V < KV, which is contrary to the hypothesis. Thus TQ does not cut the curve, and therefore it touches it. Conversely, if the tangent at a point Q meet the diameter PP' outside the section in the point T, and QV is the ordinate from Q, 'TP:'TP' = PV: VP'. Also no other straight line can fall between TQ and the curve. [This again is separately proved by Apollonius by a simple reductio ad absurdum.] * See the note on tlie previous propo^iition. 28 THE COyiCS OF APOLLONIUS. Proposition 14. [I. 37, 39.] In a hyperbola, an ellipse, or a circle, if QV be an ordinate to the diameter PP', and the tangent at Q meet PP' in T, then (1) CV.CT = CP\ (2) QF-• : CV. VT = p : PP' [or CD' : CP^]. Τ pI V C (1) Since QT is the tangent at Q, TP : ΎΡ' = PV : ΡΎ, [Prop. 13] .•. TP + TP' : TP ~ TP' = PV + P'V : PV ~ P'V- thus, for the hyperbola, 2CP:26T=26T:2CP; and for the ellipse or circle, 2CT:2GP = 2CP:2GV; therefore for all three curves CV,CT=CP\ TANGENTS. 29 (2) Since CV : CP = (T : CT. CV~ GP:CV=CP~CT: CP, Avhence PV : CV = PT : CP, or PV:PT=CV:CP. .•. PV : PV+PT = CV : CV+ CP, or PV:VT=^CV:P'V, and CV.VT=PV.P'V. But QV : PF. P'F= /) : PP' (or CD' : CP*). [Prop. 8] .•. QV : (7F. Fr = ^j : PP' (or CD» : CP'). Cor. It follows at once that QV : VT is equal to the ratio compounded of the ratios ρ : PP' (or CD' : CP') and C7 : QF. Proposition 15. [I. 38, 40.] If Qv be the ordinate to the diameter conjugate to PP', and QT, the tangent at Q, iiieet that conjugate diameter in t, then (!) Cv.Ct=CD\ (2) Qv' :Cv.vt = PP' :p [or CP' : CD'], (3) tD : tD' = vD' : vD for the hyperbola, and tD : tD' = vD : vD' for the ellipse and circle. Using the figures drawn for the preceding proposition, we have (1) QV : CV. VT = CD' : CP'. [Prop. U] But QV:CV=Cv:CV, and QV:VT=Ct:CT; .•. QV : CV. VT= Cv.Ct : CV. CT. Hence Cv . Ct : CV. CT = CD' : CP'. And CV.CT =CP'; [Pn.p. 14] .•. Cv.Ct = CD\ (2) As before, QV : CV. VT=CD' : CP' (or;) : PF). But QV : CV = Cv : Qv, 30 THE COyJC.S OF AIOLLONIUS. and QV: VT = vt :Qv; .-.QV'.CV.VT=Cv.vt:Qv'. Hence Qv' : Cv . vt = CP' : ΟΌ'' = PP' : ;). (3) Again, Ct.Cv = CD' = CD.CD': .\Ct:CD=CD' :Cv, and .•. Gt + GD : Gt~GD=GD' + Gv : GD'~Gv. Thus tD : tD' = vD' : vD for the hypevholu, and iD' : iZ) = vD' : vD for the ellipse and c?Vcie. Cor. It follows from (2) that Qv : Gv is equal to the ratio compounded of the ratios PP' : ρ (or GP^ : CZ)'^) and i/i : Qv. PROPOSITIONS LEADING TO THE REFERENCE OF A CONIC TO ANY NEW DIA:\IETER AND THE TANGENT AT ITS EXTREMITY. , atid if Proposition 16. [I. 41.] In a hyperbola, an ellipse, or a circle, if equiatir/alar paral- lelograms (VK), (PM) be described on QV, GP respectivehj, and tneir .•*» are sucK tMt |^= ^^ . § [... %. % {VN) be the parallelogram on CV similar and similarly sit η ated to (PM), then {VN)±{VK) = {PM), the lower sign applying to the hyjjerbola. Suppose to be so taken on KQ produced that QV:QO = p:PP', so that QV: QV .QO = QV : PV . PV. Thus QV.QO = PV.P'V (1). Also QV: QK = {CP : CM) . (p : PP') = (CP : CiM).{QV: QO), or (QV : QO) .{QO:QK) = (CP : CM) . (QV : QO) ; .•. QO:QK=CP:CM (2). But QO:QK=QV.QO:QV.QK and CP : CM = CP' : CP . CM : 32 THE COXICS OF APOLLONIUS. .•. CP' : GP . CM ^QV.QO'.qV.QK = PV.P'V . QV.QK, ivom i\). Therefore, since PM, VK are equiangular, GP' : PV.P'V=(PM) : (VK) (3). Hence GP' + Ρ V. P'V : GF" = {PM) + ( FZ) : {PM), Avhere the upper sign applies to the ellipse and circle and the lower to the hyperbola. and hence {VN) : {PM) = {PM) + {VK) : {PM), so that ( VN) = {PM) + { VK), or {VN)±{VK) = {PM). [The above proof is reproduced as given by ApoUonius in order to show his method of dealing with a somewhat compli- cated problem by purely geometrical means. The proposition is more shortly proved by a method more akin to algebra as follows. We have QF» : GV ~ GP' = GD' : GP\ QV_G^CP ^r. r.r. CD' GP^'CM' CD' and QK ^''■^''■CP.CM 3r QV=QK GV'~GP CP.GM' CD' : GP\ or QV.QK = GP.GM{^l'-l .■.{VK) = {VN)-{PM), {VN)±{VK) = {PM).] TRANSITION TO Λ NEW DIAMETER. 33 Proposition 17. [I. 42.] In a parabola, if QV, RW he ordinates to the diameter through P, and QT, the tangent at Q, and RU parallel to it meet the diameter in T, U respectively; and if through Q a parallel to the diameter he drawn meeting RW produced in F and the tangent at Ρ in E, then Δ R UW = the parallelogram {EW). Since QT is a tangent, TV=2PV; [Prop. 12] .•. AQTV={EV) (1). Also QV':RW' = PV:PW', .•. Δ QTV : Δ RUW={EV) : (EW), ZA and Δ QTV = (EV), from (1) ; .•. Δ RUW={EW). Proposition 18. [I. 43, 44.] In a hypei'hola, an ellipse, or a circle, if the tangent at Q and the ordinate from Q meet the diameter in T, V, and if RW he the ordinate from any point R and RU he parallel to QT ; if also RW and the parallel to it through Ρ meet CQ in F, Ε respectively, then A CFW~ A CPE= A RUW. H. C. THE ΓΌΛΥΓ.•? OF APOLLONIUS. : CV. VT = p : PP' [or CD' : OP'], PP') . {CV -.QV); [Prop. 14 and Cor.; We have QV whence QV : VT = (p therefore, by parallels, RW:WU={p: PP') . (CP : PE). Thus, by Prop. 16, the parallelograms which are the doubles of the triangles RUW, CPE, GWF have the property proved in that proposition. It follows that the same is true of the triangles themselves, .•. Δ CFW ~ Δ CPE =ARUW. [It is interesting to observe the exact significance of this proposition, which is the foundation of Apollonius' method of transformation of coordinates. The proposition amounts to this: If GP, GQ are fixed semidiameters and R a variable point, the area of the quadrilateral GFRU is constant for all positions of R on the conic. Suppose now that CP, CQ are taken as axes of coordinates {CP being the axis of a•). If we draw RX parallel to CQ to meet GP and RY parallel to CP to meet CQ, the proposition asserts that (subject to the proper convention as to sign) ARYF+CJ CXRY+ Δ RX U = {const.). But, since RX, RY, RF, BU are in fixed directions, ARYFcc RY\ or A R YF = ax- ; CJCXRY^ RX.RY, CJCXRY=βxy■, ARXlJcc RX\ ARXU= yy-. or or TRANSITION TO Λ NEW DIAMETER. 3i Heuce, if x, y are the coordinates of li, ax^ + βχι/ + ψ/ = A, which is the Cartesian equation referred to the centre as origin and any two diameters as axes.] Proposition 19. [I. 45.] If the tangent at Q and the straight line through R parallel to it meet the secondary diameter in t, ν respectively, and Qv, Rw he parallel to the diameter PP', meeting the secondary diameter in V, w ; if also Rw meet CQ inf then Δ Οβυ = Δ Ruw - Δ CQt. / ι u f A^ κ > /^V ^""^x V " \ A) .• \ \ \ [Let PK be drawn parallel to Qt meeting the secondary diameter in K, so that the triangle CPK is similar to the triangle vQti] We have [Prop. 14, Cor.] QV:CV={p.PP').{VT:QV) = {p:PP').{Qv:vt), 3—2 36 THE COXrCS OF APOLLONIUS. and the triangles QvC, Qvt are the halves of equiangular paral- lelograms on Cv (or QV) and Qv (or CV) respectively: also CPK is the triangle on CP similar to Qvt. Therefore [by Prop. 16], Δ CQv ^ A Qvt- A CPK, and clearly A CQv = A Qvt - A CQt; :.ACPK= A CQt Again, the triangle Cfw is similar to the triangle CQv, and the triangle Rwu to the triangle Qvt. Therefore, for the ordinate RW, AC/iu= A Ruw ~ A CPK = A Ruw - Δ CQt. Proposition 20. [I. 46.] In a parabola the straight line draimi through any point parallel to the diameter- bisects all cho7'ds parallel to the tangent at the point. Let RR' be any chord parallel to the tangent at Q and let it meet the diameter PF in U. Let QM drawn parallel to PF meet RR' in 31, and the straight lines drawn ordinate-wise through R, R', Ρ in F, F', Ε respectively. We have then [Prop. 17] ARUW=njEW, and AR'UW' = CJEW\ Therefore, by subtraction, the figure R W W'R' = Ο P' W. Take away the common part R'W'WFM, and we have Δ RMF= A R'MF'. And R'F' is parallel to RF; .■.RM=MR'. I TRANSITION TO A NEW DIAMETER. .37 Proposition 21. [I. 47, 48.] In a hyperbola, an ellipse, or ο circle, the line joining any point to the centre bisects the chords parallel to tlie tangent at the point κ y/ \ Ε F' ,.-''. fN I \q^ Λ R ; ~~~"~\ ; q. ^/m\ ; \ A ^ R' \ ^i A TU ~P w w \ ; If QT be the given tangent and RR' any parallel chord, let RW, R'W, Ρ Ε be drawn ordinate-wise to PP\ and let CQ meet them in F, F', Ε respectively. Further let CQ meet RR' in M. Then we have [by Prop. 18] /^CFW-^/^CPE^^tsRUW, and Δ CF' W - Δ CPE = Δ R'aW\ 38 THE comes of apollonius. Thus (1), iiu the figure is drawn for the hyperbola, ARUW = quadrilateral EPWF, and AR'U W = quadrilateral Ε Ρ W'F'; .•. , by subtraction, the figure F'W'WF= the figure R'W'WR. Taking away the common part R' W WFM, we obtain AFRM = AF'R'M. And, •.• FR, FR' are parallel, RM=MR'. (2) as the figure is drawn for the ellipse, AGPE-ACFW = ARUW, ACRE - ACFW = AR'UW, .•. , by subtraction, ACF'W - ACFW = ARUW - AR'UW, or ARUW-\- AGFW = AR'UW + ACF'W. Therefore the quadrilaterals CFRU, GF'R'U are equal, and, taking away the common part, the triangle GUM, we have AFRM=AF'R'M, and, as before, RM = MR'. (3) if RR' is a chord in the opposite branch of a hyperbola, and Q the point where QG produced meets the said opposite branch, GQ will bisect RR' provided RR' is parallel to the tangent at Q'. We have therefore to prove that the tangent at Q is parallel to the tangent at Q, and the proposition follows immediately*. * Eutocius supplies the proof of the parallelism of the two tangents as follows. We have CV.CT= CP^ [Prop. 14], and CV'.Cr = CP'^; :. cv.cT=cv'. or, and GV=GV', V i7y = Cy'[Prop. 10]; .•. CT=CT'. Hence, from the as CQT, CQ'T', it follows that QT, Q'T are parallel. TRANSITION' To A NEW DIAMETER. 30 Proposition 22. [I. 49.] Let the tangent to a parabola at F, the extremity of the ainginal diameter, meet the tangent at any point Q in 0, and the parallel through Q to the diameter in Ε ; and let RR he any chord parallel to the tangent at Q meeting PT in U and EQ produced in Μ ; then, if ρ he taken such that UQ:QE=p':2QT, it is to he proved that RM' = p'.QM. In the figure of Prop. 20 draw the ordinate Q V. Then we have, by hypothesis, 0Q:QE = p':2TQ. Also QE = PV=TP. Therefore the triangles EOQ, POT are equal. Add to each the figure QOPWF; .•. the quadrilateral QTWF= nj{EW) = Δ RUW. [Prop. 17] Subtract the quadrilateral MUWF; .•. CJQU= ARMF, and hence RM . MF = 2QM . QT (1). But RM : MF =OQ:QE = p': 2ψ\ or RM' : RM . MF = p' . QM : 2QM . QT. Therefore, from (1), RM' = ρ . QM. Proposition 23. [I. 50.] If in a hyperhola, an ellipse, or a circle, the tangents at P, Q meet in 0, and the tangent at Ρ meet the line joining Q to the centre in Ε ; if also a length QL (= p) he taken such that OQ : QE = QL : 2TQ 40 THE COSICS OV Al'ULLONlUS. and erected perpendicular to QC ; if further Q'L be joined {wJiere Q' is on QC produced and CQ= CQ'), and MK he drawn parallel to QL to meet Q'L in Κ (where Μ is the point of concourse of CQ and RR, a chord parallel to the tangent at Q): then it is to he proved that RM' = QM.MK. In the figures of Prop. 21 draw CHN parallel to QL, meet- ing QL in Η and MK in N, and let ii! W be an ordinate to PP', meeting CQ in F. Then, since CQ = CQ\ QH = HL. Also 0Q:QE = QL:2QT = QH:QT; .•. RM:MF=QH:QT (A). Now /\RUW = /\GFW-AGPE = l^CFW~liCQT'') .'.in the figures as drawn (1) for the hyperbola, ARUW=QTWF, .•. , subtracting 3IUWF, •we have ARMF=QTUM. (2) for the ellipse and circle, ARUW = ACQT-AGFW; .•. Δ CQT= quadrilateral /e UCF; and, subtracting A MUG, we have ARMF=QTUM. RM.MF=QM{QT+MU) (B). * It will be observed that Apollonius here assumes the equality of the two triangles CPE, CQT, though it is not until Prop. 53 [III. 1] that this equality is actually proved. But Eutocius gives another proof of Prop. 18 which, he says, appears in some copies, and which begins by proving these two triangles to be equal by exactly the same method as is used in our text of the later proof. If then the alternative proof is genuine, we have an explanation of the assumption here. If not, we should be tempted to suppose that Apollonius quoted the property as an obvious limiting case of Prop. 18 [I. 43, 44] where II coincides with Q ; but this would be contrary to the usual practice of Greek geometers who, no doubt for tlie purpose of securing greater stringency, preferred to give separate proofs of tlie limiting cases, though the parallelism of the respective proofs suggests that they were not unaware of the connexion between the general theorem and its limiting cases. Compare Prop. 81 [V. 2], where Apollonius proves separately the case where Ρ coincides with B, though we have for tlie sake of brevity only mentioned it as a limiting case. TRANSITION TO A NEW DIAMKTKR. 41 Now QT : MU= CQ:GM=QH: MN, .•.QH + ^fN : QT + MU= QH : QT = RM : MF [from (A)] ; .•. QM{QH + MN) : QM{QT+MU) = RM' : RM.MF; .•. [by (B)] RM* = QM(QH + MN) = QM.MK. The same is true for the opposite branch of the hyperbola. The tangent at Q' is parallel to QT, and P'E' to PE. [Prop. 21, Note.] .•. O'Q' : Q'E' =OQ:QE=p' : 2QT=p' : 2Q'r, whence the proposition follows. It results from the propositions just proved that in a parabola all straight lines drawn parallel to the original diameter are diameters, and in the hyperbola and ellipse all straight lines drawn through the centre are diameters ; also that the conies can each be referred indiiferently to any diameter and the tangent at its extremity as axes. CONSTRUCTION OF CONICS FROM CERTAIN DATA. Proposition 24. (Problem.) [I. 52, 53.] Given a straight line in a fixed plane and terminating in a fi^ed point, and another straight line of a certain length, to find a parabola in the plane such that the first straight line is a diameter, the second straight line is the corresponding parameter, and the ordinates are inclined to the diameter at a given angle. First, let the given angle be a right angle, so that the given straight line is to be the axis. Let AB be the given straight line terminating at A, pa the given length. Produce Β A to C so that AC > —^ , and let S be a mean 4 proportional between AG and pa- (Thus pa : AC = S' : AG^, and AC>lpa, Avhence AC'^ > -τ- , or 2AG > S, so that it is possible to describe an isosceles triangle having two sides equal to AG and the third equal to S.) Let AUG be an isosceles triangle in a plane perpendicular to the given plane and such that AO = AG, DC = S. Complete the parallelogram AGOE, and about A Ε as diameter, in a plane perpendicular to that of the triangle AUG, describe a circle, and let a cone be drawn with as PROBLEMS. 43 apex and the said circle as base. Then the cone is a right cone because OE = AG = OA. Produce OE, OA to H, K, and draw Η Κ parallel to AE, and let the cone be cut by a plane through HK parallel to the base of the cone. This plane will produce a circular section, and will hitcrscct the original plane in a line PP', cutting AB at right angles in N. Now Pa•. AE = AE: AO, since AE= 00== S,AO = AC; .-. pa:AO = AE':AO' = AE':AO.OE. Hence PAP' is a parabola in which ;;„ is the parameter of the ordinates to AB. [Prop. 1] Secondly, let the given angle not be right. Let the line which is to be the diameter be PM, let ρ be the length of the parameter, and let MP be produced to F so that PF = ^p. Make the angle FPT equal to the given angle and draw FT perpendicidar to TP. Draw TiV parallel to PM, and PN perpen- dicular to TN; bisect TN in A and draw LAE through A perpendicular to FP meeting PT in ; and let NA.AL = PN\ Now with axis AN and parameter AL describe a para- bola, as in the first case. This will pass through Ρ since PN^ = LA . AN. Also PT will be a tangent to it since AT = AN. And PM is parallel to AN. Therefore PM is a dia- meter of the parabola bisecting chords parallel to the tangent PT, which are therefore inclined to the diameter at the given angle. Again the triangles FTP, OEP are similar : ..OP:PE=FP:PT, = p:-2PT, by hypothesis. Therefore ρ is the parameter of tht the diameter PM. [Prop. 22] parabola corresponding to 44 THE COXICS OF APOLLONIUS, Proposition 25. (Problem.) [I. 54, 55, 59.] Giveti a straight line AA' in a plane, and also another straight line of a certain length; to find a hyperbola in the plane such that the first straight line is a diameter of it and the second equal to the corresponding parameter, while the ordinates to the diameter make with it a given angle. First, let the given angle be a ngJit angle. Let AA', Pa be the given straight lines, and let a circle be drawn through A, A' in a plane pei-pendicular to the given plane and such that, if G be the middle point of A A' and DF the diameter perpendicular to A A ' , DC '. CF 1sr AA' '. Pa. Then, if BC : CF = A A' : pa, we should use the point F for our construction, but, if not, suppose DC:GG = AA':pa (GG being less than GF). Draw GO parallel to AA', meeting the circle in 0. Join AG, A'O, DO. Draw AE parallel to DO meeting A'O produced in E. Let DO meet A A' in B, PROBLEMS. 45 Then Z0EA = ZAOD= ζ AnD=zOAE: .•. OA = OE. Let a cone be described with for apex and for base the circle whose diameter ϊά AE and whose plane is perpendicular to that of the circle AOD. The cone will therefore be right, since OA = OE. Produce OE, OA to //, Κ and draw Η Κ parallel to AE. Draw a plane through HK perpendicular to the plane of the circle AOD. This plane will be parallel to the base of the cone, and the resulting section Avill be a circle cutting the original plane in PP' at right angles to A' A produced. Let GO meet HK in M. Then, because Ν A meets HO produced beyond 0, the curve PAP' is a hyperbola. And AA':pa = DC:CG = DB:BO = ΌΒ.Β0:Β0' = A'B.BA :B0\ But A'B : BO = OM : MH] , . ., . , BA:BO = OM : ΜΚί ^^ '''''^'^' '"'''"^^"'• .•. A'B. ΒΑ : BO'=OIiP : HM.MK. Hence AA' : pa= OM' : HM . MK. Therefore pa, is the parameter of the hyperbola PAP' cor- responding to the diameter AA'. [Prop. 2] Secondly, let the given angle not be a right angle. Let PP', ρ be the given straight lines, OPT the given angle, and C the middle point of PP'. On CP describe a semicircle, and let Ν be such a point on it that, if NH is drawn parallel to PT to meet CP produced in H, NH':CH.HP=p:PP'*. * This conetruction is assumed by Apollonius without any explanation ; but we may infer that it was aiTived at by a method simihir to that adopted for 46 THE CONICS OF APOLLONIUS. Join NO meeting PT in T, and take A on CN such that CA^=CT. CN. Join PiY and produce it to Κ so that ΡΝ' = Λλ^.ΝΚ. Produce AC to A' so that AC = CA', join A'K, and draw EOAM through A parallel to PN meeting CP, ΡΓ, A'K in -£^, 0, Jlf respectively. With AA' as axis, and AM as the corresponding parameter, describe a hyperbola as in the first part of the proposition. This will pass through Ρ because PN^ = AN .NK. a similar case in Prop. 52. In fact the solution given by Eutocius represents sufficiently closely Apollonius' probable procedure. If HN produced be supposed to meet the curve again in Λ", then N'H.HN=CH.HP; :. Nm : CH.HP = NH : N'H. Thus we have to draw HNN' at a given inclination to PC and so that N'H:NH = PP' : p. Take any straight line o/3 and divide it at 7 so that aβ■.βy = PP':p. Bisect 07 in δ. Then draAV from G, the centre of the semicircle, GR at right angles to PT which is in the given direction, and let GR meet the circumference in R. Then RF drawn parallel to PT will be the tangent at R. Suppose RF meets CP produced in F. Divide FR at .S' so that FS : SR — βy : y8, and produce FR to S" so that RS' = RS. Join GS, GS', meeting the semicircle in N, N', and join N'N and produce it to meet CF in H. Then Nil is the straight line which it was required to find. The proof is obvious. PROBLEMS. 47 Also PT Λνΐΐΐ be the tangent at Ρ because CT.CN=CA\ Therefore CP will be a diameter of the hyperbola bisecting chords parallel to PT and therefore inclined to the diameter at the given angle. Again we have ρ : 2CP = NH' : CH . HP, by construction, and 2CP : 2PT = GH : NH ^GH.HP.NH.HP; .\ρ•ΛΡΤ = ΝΗ•'•.ΝΗ.ΗΡ = Ν Η : HP = OP : ΡΕ, by similar triangles ; therefore ρ is the parameter corresponding to the diameter PP'. [Prop. 23] The opposite branch of the hyperbola with vertex A' can be described in the same way. Proposition 26. (Problem.) [I. 60.] Criven Ηυο straight lines bisecting one another at any angle, to describe two hyperbolas each with two branches such that the straight lines are conjugate diameter's of both hyperbolas. Let PP', DD' be the two straight lines bisecting each other at α 48 THE OOXTCS f)F APOLLONIUS. From Ρ draw PL perpendicular to PP" and of such a length that PP' . PL = DD"' ; then, as in Prop. 25, describe a double hyperbola with diameter PP' and parameter PL and such that the ordinates in it to PP' are parallel to DD'. Then PP', DD' are conjugate diameters of the hyperbola so constructed. Again, draw DM perpendicular to DD' of such a length that DM . DD' = PP'^ ; and, with DD' as diameter, and DM as the corresponding parameter, describe a double hyperbola such that the ordinates in it to DD' are parallel to PP'. Then DD', PP' are conjugate diameters to this hyperbola, and DD' is the transverse, while PP' is the secondary dia- meter. The two hyperbolas so constructed are called conjugate hyperbolas, and that last dra\vn is the hyperbola conjugate to the first. Proposition 27. (Problem.) [I. 56, 57, 58.] Given a diameter of an ellipse, the corresponding parameter, and the angle of inclination between the diameter and its ordi- nates : to find the ellipse. First, let the angle of inclination be a right angle, and let the diameter be greater than its parameter. PROBLEMS. 49 Let ΛΑ' he the diameter and AL, ά straight line of length Pa perpendicular to it, the parameter. In a plane at right angles to the plane containing the diameter and parameter describe a segment of a circle on AA' as base. Take AD on A A' equal to AL. Draw A E, A'E to meet at E, the middle point of the segment. Draw DF parallel to A'E meeting A Ε in F, and OFN parallel to A A' meeting the circumference in 0. Join EO and produce it to meet A'A produced in T. Through any point Η on OA produced draw HKMN parallel to OE meeting OA', AA', OF in K, M, Ν respectively. ΝοΛν Ζ TO A = ζ OEA + ζ OAE = ζ AA'O + ^ OA'E ■- = δΕΑΑ'= δΕΟΑ', and HK is parallel to OE, whence Ζ OH Κ = Ζ OKH, Ζ ΑΑΈ and OH=OK. Η. C. 50 THE COXICS OF APOLLONIUS. With as vertex, and as base the circle draAvn with diameter HK and in a plane perpendicular to that of the triangle OHK, let a cone be described. This cone λυΙΙΙ be a right cone because OH = OK. Consider the section of this cone by the plane containing AA', AL. This will be an ellipse. And Pn Now AA' = AD : = AF: = TO : = T0': TA = HN and AA' AE TE '.TO. Τ Ε ; ΤΑ . ΤΑ'. TO.TA = ΗΝ : NO, TO :ΤΑ' = Ν Κ : NO, by similar triangles, TA.TA' = HN.NK:NO\ j)a:AA' = HN.NK:NO\ or Pa is the parameter of the ordinates to AA'. [Prop. 3] Secondly, if the angle of inclination of the ordinates be still a right angle, but the given diameter less than the para- meter, let them be BB', BM respectively. Let C be the middle point οι ΒΒ',Άπά through it draw^^', perpendicular to BB' and bisected at C, such that TO' that AA" = BB'.BM: and draw AL, parallel to BB', such that BM : BB' = AA' : AL thus A A' > AL. PllOHLEMS. 51 Now with ΛΛ' as diameter and AL as the corresponding parameter describe an ellipse in which the ordinates to ΛΛ' are perpendicular to it, as above. This will be the ellipse required, for (1) it passes through B, B' because AL : AA' = BB' : BM = BB" : AA" = BC":AC.CA', (2) BM : BB' = AC' : BC = AC':BC.CB', so that BM is the parameter corresponding to BB'. Thirdly, let the given angle not be a right angle but equal to the angle CPT, where G is the middle point of the given diameter PP' ; and let PL be the parameter coiTCspond- ing to PP'. Take a point N, on the semicircle which has CP for its diameter, such that NH drawn parallel to PT satisfies the relation NH' : CH.HP = PL : PP'*. * This construction like that in Prop. 25 is assumed \vithont explanation. If NH be supposed to meet the other semicircle on CP as diameter in N', the 4—2 ό2 THE COXICS OF APOLLONIUS. Join CN and produce it to meet PT in T. Take Λ, on CT, such that GT.CN = CA\ and produce AG to A' so that AG = CA'. Join PiV and produce it to Κ so that AN'.NK = PN\ Join -4'ir. Draw Ε AM through A perpendicular to CA (and therefore parallel to NK) meeting GP produced in E, PT in 0, and A' Κ produced in M. Then with axis A A' and parameter AM describe an ellipse as in the first part of this proposition. This will be the ellipse required. For (1) it will pass through Ρ •.• PN' = AN.NK. For a similar reason, it will pass through P' •.• GP' = GP and GA' = GA. (2) PT will be the tangent at Ρ •.• GT . GN=GA\ (3) We have ]3 : 2CP = NH^ : GH . HP, and 2GP : 2PT = GH : HN = GH.HP :NH.HP; .•. ex aequali ρ : 2PT = NW : NH . HP = NH:HP = OP : PE. Therefore ρ is the parameter corresponding to PP'. [Prop. 23] problem here reduces to drawing NHN' in a given direction (parallel to PT) so that N'H:NH = PP':p, and tiie construction can be effected by the method shown in the note to Prop. 25 mutatis mutandis. ASYMPTOTES. Proposition 28. [IL 1, 15, 17, 21.] (1) If PP' he a diameter of a hyperbola and ρ the corre- sponding parameter, and if on the tangent at Ρ there he set off on each side equal lengths PL, PL', such that PU = PL" = ip . PP' [= GB'l then CL, CL' produced will not meet the curve in any finite point and are accordingly defined as asymptotes. (2) The opposite branches have the same asymptotes. (3) Conjugate hyperbolas have their asymptotes common. (1) If possible, let CL meet the hyperbola in Q. Draw the ordinate QV, which will accordingly be parallel to LU, Now p. PP'=p. PP' : PP"' = PL' : CP•' = QV':GV\ 54ί THE COXICS OF APOLLONIUS. But p:PP' = QV':PV.P'V. ... PV.P'V=CV\ i.e. CV - CP' = C]^, which is absurd. Therefore GL does not meet the hyperbola in any finite point, and the same is true for CL'. In other words, GL, GL' are asymptotes. (2) If the tangent at P' (on the opposite branch) be taken, and P'M, P'M' measured on it such that P'M' = P'M" = CD\ it folloAvs in like manner that GM, GM' are asymptotes. Now MM', LL' are parallel, PL = P'M, and PGP' is a straight line. Therefore LGM is a straight line. So also is L'GM', and therefore the opposite branches have the same asymptotes. (3) Let PP', DD' be conjugate diameters of tAvo conjugate hyperbolas. Draw the tangents at P, P, D, U. Then [Prop. 11 and Prop. 26] the tangents form a parallelogram, and the diagonals of it, LM, L'M', pass through the centre. Also PL = PL' = P'M = P'M' = GD. Therefore LM, L'M' are the asymptotes of the hyperbola in which PP' is a transverse diameter and DD' its conjugate. Similarly DL = DM' = D'L' = D'M= GP, and LM, L'M' are the asymptotes of the hyperbola in which DD' is a transverse diameter and PP' its conjugate, i.e. the conjugate hyperbola. Therefore conjugate hyperbolas have their asymptotes common. ASYMPTOTES. Proposition 29. [II. 2.] No straight line through G luithin the angle between the asymptotes can itself he an asymptote. If possible, let CK be an asymptote. Draw from Ρ the straight line PK parallel to GL and meeting GK in K, and through Κ draw BKQR parallel to LL', the tangent at P. Then, since PL = PL', and RR, LL' are parallel, iiF= R'V, where V is the point of intersection of RR and GP. And, since PKRL is a parallelogram, PK = LR, PL = KR. Therefore QR > PL. AhoRQ>PL'; .•. RQ.QR'>PL.PL', or ΡΓ (1). Again and thus whence RV"- P GV' = PU : GP'=p:PP', PP' = QV'.PV.P'V = QV':GV'-GP': GV' = QV':GV'-GP' --^RV'-QV: GP'; ■.GP' = RV'- QV':GP\ PL'=RV'-QV'=RQ.QR', which is impossible, by (1) above. Therefore GK cannot be an asymptote. [Prop. 28] [Prop. H] RV PL' 56 THE COXICS OF APOLLONIUS. Proposition 30. [11. 3.] If a straight line touch a hyperbola at P, it will meet the asymptotes in two points L, L' ; LL' luill he bisected at P, and Pr = ip.PP'[=GD']. [This proposition is the converse of Prop. 28 (1) above.] For, if the tangent at Ρ does not meet the asymptotes in the points L, L' described, take on the tangent lengths PK, PK' each equal to CD. Then GK, GK' are asymptotes ; which is impossible. Therefore the points K, K' must be identical with the points L, L' on the asymptotes. Proposition 31. (Problem.) [11. 4.] Given the asymptotes and a point Ρ on a hyperbola, to find the curve. Let GL, GL' be the asymptotes, and Ρ the point. Produce PG to P' so that GP=GP'. Draw PK parallel to GL' meeting GL in K, and let GL be made equal to twice GK. Join LP and produce it to L'. Take a length ρ such that LL'^ =p.PP', and with diameter PP' and parameter ρ describe a hyperbola such that the ordinatcs to PP' arc parallel to /.//. [Prop. 25] ASYMPTOTES. 57 Proposition 32. [II. 8, 10.] If Qq be any chord, it will, if produced both ^vays, meet the asymptotes in two points as R, r, and (1) QR, qr will ι (2) RQ.Qr = lp.PP'[=CD'l Tako V the middle point of Qq, and join CV meeting the curve in P. Then CF is a diameter and the tangent at Ρ is parallel to Qq. [Prop. 11] Also the tangent at Ρ meets the asymptotes (in L, L'). Therefore Qq parallel to it also meets the asymptotes. Then (1), since Qq is parallel to LL', and LP = PL', it follows that RV th( But 3reforc i, subtracting QV- QR- -Vq; = qr. (2) We have p:PF = = = PL' -.RV CP' an d ρ : PP' = --QV CV- OP .PL':CP' = --p:PP' = --RV -QV: CP = --RQ.Qr:CP' ; th LIS RQ.Qr= --PL' — --\p.PP' = CD\ Similarly y rq.qR^ -- CD\ [Prop. 8] 58 THE COXICS OF APOLLONIUS. Proposition 33. [II. 11, 16.] If Q, Q are on opposite branches, and QQ' meet the asi/7)ip- totes in K, K', and if CF be the seniidianieter parallel to QQ', then (1) KQ.QK' = CP\ (2) QK=Q'K'. Draw the tangent at Ρ meeting the asymptotes in L, L', and let the chord Qq parallel to LL' meet the asymptote.s in R, r. Qq is therefore a double ordinate to CP. Then we have Ρ Γ : CP' = (PL : CP) . (PL' : CP) = (RQ:KQ).(Qr:Q]r) = RQ.Qr:KQ.QK'. Pr==RQ.Qr; •.KQ.QK' = CP\ K'Q'.Q'K=CP\ KQ . QK' = CP' = K'Q' . Q'K ; .•. KQ . {KQ + KK') = K'QXK'Q' + KK'), whence it follows that KQ = Λ''^'. But Similarly (2) [Prop. 32] ASYMFrOTES. 59 Proposition 34. [IT. 12.] If Q, q he any two points on a hyperbola, and parallel straight lines QH, qh be drawn to meet one asymptote at any angle, and QK, qk {also parallel to one another) meet the other asymptote at any angle, then HQ . QK = hq. qk. Let Qq meet the asymptotes in R, r. We have liQ .Qr = Rq .qr; .•. RQ : Rq = qr : Qr. But RQ : Rq = HQ : hq, and qr : Qr = qk : QK ; .•. HQ : hq = qk : QK, or HQ . QK = hq . qk. [Prop. 82] 60 THE COXICS OF APOLLONIUS, Proposition 35. [II. 13.] //' in the space between the asymptotes and the hyperbola a straight line be drawn parallel to one of the asymptotes, it will meet the hyperbola in one point only. Let .£^ be a point on one asymptote, and let EF be drawn parallel to the other. Then EF produced shall meet the curve in one point only. For, if possible, let it not meet the curve. Take Q, any point on the curve, and draAv QH, QK each parallel to one asymptote and meeting the other ; let a point F be taken on EF such that HQ.QK=CE.EF. Join OF and produce it to meet the curve in q ; and draw qh, qk respectively parallel to QH, QK. Then hq.qk = HQ. QK, [Prop. 34] and HQ.QK=CE. EF, by hypothesis, :.hq.qk=GE.EF: which is impossible, •.• hq > EF, and qk > CE. Therefore EF will meet the hyperbola in one point, as R. Again, EF will not meet the hyperbola in any other point. For, if possible, let EF meet it in R' as well as R, and let RM, R'M' be drawn parallel to QK. Then ER . RM = ER' . R'M' : [Prop. 34] which is impossible, •.• ER' > ER. Therefore EF does not meet the hyperbola in a second point R'. ASYMPTOTES. 61 Proposition 36. [II. 14] The asymptotes and the hyperbola, as they pass on to infinity, approach continually nearer, and will come within a distance less than any assignable length. Let S be the given length. Draw two parallel chords Qq, Q'q' meeting the asyntiptotes in li, r and R', ?•'. Join Cq and produce it to meet Q'q' in F. κ Then r'q' . q'R = rq . qK, and q'R > qR ; .•. q'r' < qr, and hence, as successive chords are taken more and more distant from the centre, qr becomes smaller and smaller. Take now on rq a length rH less than S, and draw II^f parallel to the asymptote Cr. HM will then meet the curve [Prop. 35] in a point M. And, if MK be drawn parallel to Qq to meet Cr in K, Μ Κ = rH, whence MK < S. 62 THE COXICS OF APOLLONIUS. Proposition 37. [II. 19.] Any tangent to the conjugate hyperbola luill meet both branches of the original hyperbola and be bisected at the point of contact. (1) Let a tangent be drawn to either branch of the conju- gate hyperbola at a point D. This tangent will then meet the asymptotes [Prop. 30], and will therefore meet both branches of the original hyperbola. (2) Let the tangent meet the asymptotes in L, Μ and the original hj^perbola in Q, Q. Then [Prop. 30] DL = DM. Also [Prop. 33] LQ = MQ' ; whence, by addition, DQ = !>(/. ASYMPTOTES. 63 Proposition 38. [11. 28.] If a cJiord Qq in one branch of a hyperbola meet the asymp- totes in R, r and the conjugate hyperbola in Q', q, then Q'Q.Qq'=2GD\ Let CD be the parallel semi-diamctcr. Then we have [Props. 32, 33] RQ.Qr=CD\ RQ'.qr=CD'; .'. 2CD' = RQ . Qr + RQ' . Φ' = (RQ + RQ')Qr + RQ'.QQ' = QQ'.{Qr + RQ') -=QQ'(Qr + rq') = QQ'.Qq. TANGENTS, CONJUGATE DIAMETERS AND AXES. Proposition 39. [II. 20.] If Q he any point on a hyperbola, and CE he drawn from the centre parallel to the tangent at Q to meet the conjugate hyperhola in E, then (1) the tangent at Ε will he parallel to CQ, and (2) CQ, GE will he conjugate diameters. Let FP', DD' be the conjugate diameters of reference, and let QF be the ordinate from Q to PP', and EW the ordinate from Ε to DD' . Let the tangent at Q meet PP', DD' in T, t respectively, let the tangent at Ε meet DD' in U, and let the tangent at D meet EU, CE in 0, Η respectively. Let p, p' be the parameters corresponding to PP', DD' in the two hyperbolas, and we have (1) PP' :p=p' :DD', [■.p. PP' = DD'\ p' . DD' = PP''] TANGENTS, CONJUGATE DIAMETERS AND AXES. 60 and PP' ■.p = CV.VT: QV\ ρ : OD' = EW : GW . WU. [Prop. 14] .•. CV.VT.QV"- = EW : CW . WU. But, by similar triangles, VT:QV=EW.GW. Therefore, by division, CV:QV = EW: WU. And in the triangles CVQ, EWU the angles at V, W are equal. Therefore the triangles are similar, and ^QCV= ZUEW. But ζ VCE = ζ CEW, since EW, OFare parallel. Therefore, by subtraction, Ζ QCE = Ζ CEU Hence EU is parallel to CQ. (2) Take a straight line S of such length that HE:EO = EU : S, so that *S' is equal to half the parameter of the ordinates to the diameter EE' of the conjugate hyperbola. [Prop. 23] Also Ct.QV= GD\ (since QV = Cv), or Ct:QV=Gf:CD\ Now Ct ■.QV=tT:TQ=AtCT: ACQT, and Ce :GD'= A tCT : Δ CDH = AtCT : ACEU [as in Prop. 28]. It follows that AGQT= ACEU And zCQT=zCEU. .•. CQ.QT=CE.EU (A). But S:EU=OE:EH = CQ : QT. .•. S.CE : CE.EU=CQ' -.CQ.QT. Hence, by (A), S.CE=CQ\ .•. 2S.EE' = QQ'\ where 2S is the parameter corresponding to EE'. And similarly it may be proved that EE'^ is equal to the rectangle contained by QQ' and the corresponding parameter. Therefore QQ', EE' are conjugate diameters. [Prop. 26] H. c. ') 66 THE COXICS OF APOLLONIUS. Proposition 40. [II. 87.] Jf Q, Q' cij-e any points on opposite branches, and ν the middle point of the chord QC/, then Cv is the 'secondary" diameter corresponding to the transverse diameter draiun parallel to QQ'. Join Q'C and produce it to meet the hyperbola in q. Join Qq, and draw the diameter PP' parallel to QQ'. Then we have CQ' = Cq, and Q'v = Qv. Therefore Qq is parallel to Cv. Let the diameter PP' produced meet Qq in V. Now QV=Cv=Vq, because CQ' = Cq. Therefore the ordinates to PP' are parallel to Qq, and therefore to Cv. Hence PP', Cv are conjugate diameters. [Prop. 6] Proposition 41. [II. 29, 80, 88.] // two tangents TQ, TQ' he drawn to a conic, and V he the middle point of the chord of contact QQ', then TV is a diameter. For, if not, let VE be a diameter, meeting TQ' in E. Join EQ meetiug the curve in R, and draw the chord RR' parallel to QQ' meeting EV, EQ' respectively in K, H. Then, .since RH is parallel to QQ', and QV=Q'V, RK = KH. TANGENTS, CONJLTOATE DIAMETERS AND AXES. Also, since RR' is a chord parallel to QQ' bisected by the diameter EV, RK = KR'. Therefore KR' = KH : which is impossible. Therefore EY is not a diameter, and it may be proved in like manner that no other straight line through F is a diameter except TV. Conversely, the diameter of the conic draiun through T, the point of intersection of the tangents, luill bisect the chord of contact QQ'. [This is separately proved by Apollonius by means of an easy rediictio ad absiirdum.] Proposition 42. [II. 40.] If tQ, tQ' be tangents to opposite branches of a hyperbola, and a chord RR' be drawn through t parallel to QQ', then the lines joining R, R' to v, the middle point of QQ', will be tangents at R, R'. 68 THE CONICS OF APOLLONIUS. Join vt. vt is then the diameter conjugate to the transverse diameter drawn parallel to QQ', i.e. to PP'. But, since the tangent Qt meets the secondary diameter in t, Cv . a = Ip . PP' [= CD']. [Prop. 15] Therefore the relation between ν and t is reciprocal, and the tangents &t R, R' intersect in v. Proposition 43. [II. 26, 4], 42.] In a conic, or a circle, or in conjugate hyperbolas, if two chords not passing through the centre intersect, they do not bisect each other. Let Qq, Rr, two chords not passing through the centre, meet in 0. Join CO, and draw the diameters Pj>, P'p' re- spectively parallel to Qq, Rr. Then Qq, Rr shall not bisect one another. For, if possible, let each be bisected in 0. TANGENTS, CONJUGATE DIAMETERS AND AXKS. (iU Then, since Qq is bisected in and Pp is a diameter parallel to it, CO, Fp are conjugate diameters. Therefore the tangent at Ρ is parallel to GO. Similarly it can be proved that the tangent at P' is parallel to CO. Therefore the tangents at P, P' are parallel : which is impossible, since PP' is not a diameter. Therefore Qq, Rr do not bisect one another. Proposition 44. (Problem.) [II. 44, 45.] To find a diameter of a conic, and the centre of a central conic. (1) Draw two parallel chords and join their middle points. The joining line will then be a diameter. (2) Draw any two diameters ; and these will meet in, and so determine, the centre. Proposition 45. (Problem.) [II. 4G, 47.] To find the axis of a parabola, and the axes of a central ic. (1) In the case of the parabola, let PD be any diameter. Draw any chord QQ' perpendicular to PD, and let Ν be its middle point. Then AN drawn thr(jugh Ν parallel to PD will be the axis. For, being parallel to PD, J.iVis a diameter, and, inasmuch as it bisects QQ' at right angles, it is the a.xis. And there is only one axis because there is only one diameter which bisects QQ'. V^ 70 THE COSICS OF APOLLONIUS. (2) In the Ccose of a central conic, take any point Ρ on the conic, and with centre C and radius CP describe a circle cutting the conic in P, P', Q', Q. Let PP', PQ be two common chords not passing through the centre, and let iV, 31 be their middle points respectively. Join CN, CM. Then ON, CM will both be axes because they are both diameters bisecting chords at right angles. They are also conjugate because each bisects chords parallel to the other. Proposition 46. [II. 48.] No central conic has more than two axes. If possible, let there be another axis GL. Through P' draw P'L perpendicular to CL, and produce P'L to meet the curve again in R. Join CP, CM. TANGENTS, CONJUGATE DIAMETERS AND AXES. 71 Then, since CL is an axis, PL = LR\ therefore also CP =CP' = CR. Now in the case of the ht/perhola it is clear that the circle PP' cannot meet the same branch of the hyperbola in any other points than P, P'. Therefore the assumption is absurd. In the ellipse draw RK, PH perpendicular to the (minor) axis which is parallel to PP'. Then, since it was proved that CP = CR, CP' = CR\ or CH' + HP' = CK' + KR \ .\CK'-CH' = HP'-KR' (1). Now BK.KB' + CK' = CB \ and BH.HB' + CH'=CB\ .•. CK' - CH' = BH . HB' - BK . KB'. Hence HP' - KR' = HH . HB' - BK . KB', from (1). But, since PH, RK are ordinates to BB', PH' : BH. HB' = RK' : BK.KB', and the difference between the antecedents has been proved equal to the difference between the consequents. .'.PH' = BH.HB', and RK'=- BK.KB'. .•. P, R are points on a circle with diameter BB' : which is absurd. Hence CL is not an axis. 72 THE COXICS OF APOLLONIUS. Proposition 47. (Problem.) [II. 49.] To draw a tangent to a parabola through any point on or outside the curve. (1) Let the point be Ρ on the curve. DraAv Ρ Ν per- peudicular to the axis, and produce Ν A to Τ so that AT = AN. Joiu PT Then, since AT=AN, PT is the tangent at P. [Prop. 12] In the particular case where Ρ coincides with A, the vertex, the perpendicular to the axis through A is the tangent. (2) Let the given point be any external point 0. Draw the diameter OBV meeting the curve at B, and make BV ecpial to OB. Then draw through V the straight line VP parallel to the tangent at Β [drawn as in (1)] meeting the curve in P. Join OP. OP is the tangent requii'cd, because PV, being parallel to the tangent at B, is an ordinate to BV, and OB = BV. [Prop. 12] [This construction obviously gives the two tangents through 0.] TANGENTS, CONJUGATE DIAMETERS AND AXES. 73 Proposition 48. (Problem.) [II. 49.] To draiu a tangent to a hyperbola through any point on or outside the curve. There are here four cases. Case I. Let the point be Q ou the curve. Draw QN perpendicular to the axis A A' produced, and take on A A' a point Τ such that A'T -. AT = A'N : AN. Join TQ. Then TQ is the tangent at Q. [Prop. 13] In the particular case where Q coincides with A or A' the perpendicular to the axis at that point is the tangent. Case II. Let the point be any point within the angle contained by the asymptotes. Join CO and produce it both ways to meet the hyperbola in P, P'. Take a point V on CP produced such that P'V:PV=OP': OP, and through V draw VQ parallel to the tangent at Ρ [drawn as in Case I.] meeting the curve in Q. Join OQ. Then, since QF is parallel to the tangent at P, QV \s an ordinate to the diameter P'P, and moreover P'V:PV=OP' : OP. Therefore OQ is the tangent at Q. [Prop. 13] [This construction obviously gives the two tangents through 0.] 74 THE COMCS OF Al'OLLUNlU.S. Case III. Let the point (J be on one of the asymptotes. Bisect CO at H, and through Η draw HP parallel to the other asymptote meeting the curve in P, Join OP and produce it to meet the other asymptote in L. Then, by parallels, OP : PL = OH : HC, whence OP = PL. Therefore OL touches the hyperbola at P. [Props. 28, 30] Case IV. Let the point lie within one of the exterior angles made by the asymptotes. Join CO. Take any chord Qq parallel to CO, and let V be its middle point. Draw through V the diameter PP'. Then PP' is the diameter conjugate to CO. Now take on OC produced a point w such that CO . Cw = ^p . PP' [= C'Z)*], and draAv through w the straight line wR pai-allel to PP' meeting the curve in li. Join OR. Then, since Rw is parallel to CP and Ciu conjugate to it, while CO . Cw = CD^, OR is the tangent at R. [Prop. 15] TANGENTS, CONJUGATE DIAMETERS AND AXES. 75 Proposition 49. (Problem.) [II. 49.] To draw a tangent to an ellipse through any point on or outside the curve. There are here two cases, (1) where the point is on the curve, and (2) where it is outside the curve ; and the con- structions correspond, mutatis mutandis, with Cases I. and II. of the h^'perbola just given, depending as before on Prop. 13. When the point is external to the ellipse, the construction gives, as before, the two tangents through the point. Proposition 50. (Problem.) [II. 50.] To draw a tangent to a given conic making with the auis an angle equal to a given acute angle. I. Let the conic be a parabola, and let DEF be the given acute angle. Draw DF perpendicular to EF, bisect EF at H, and join DH. Now let AN be the axis of the parabola, and make the angle NAP ecjual to the angle DHF. Let AP meet the curve in P. Draw Ρ Ν perpendicular to AN. Produce Ν A to Τ so that AN = AT, and join PT. Then PT is a tangent, and wc have to prove that ΔΡΤΝ = ΔϋΕΡ. 76 THE cogues OF APOLLONIUS. Since zDHF = zFAN, UF:FD = AN:NP. .•. 2HF.FD = 2AN:NF, or EF : FD = TiY : NF. .•.zFTN = zDEF. II. Let the conic be a central conic. Then, for the hyperbola, it is a necessary condition of the possibility of the solution that the given angle DEF must be gi'cater than the angle botAveen the axis and an asymptote, or half that between the asymptotes. If DEF be the given angle and DF be at right angles to EF, let Η be so taken on DF that Ζ HEF=zACZ, or half the angle between the asymptotes. Let A Ζ he the tangent at A meeting an asympt(jte in Z. TANGENTS, CONJUGATE DIAMETERS AND AXES. 77 \Vc have then CA^ : AZ' (or CA' : CfB') = EF' : FH\ ..CA': CB' > EF' : FJ)\ Take a point Κ on FE produced such that CA':CB' = KF.FE: FD\ Thus KF':FD^>CA':AZ\ Therefore, if DK be joined, the angle DKF is less than the angle ACZ. Hence, if the angle ΑΛ!Ρ be made equal to the angle DKF, CP must meet the hyperbola in some point P. In the case of the ellipse Κ has to be taken on EF produced so that CA- : CB' = KF .FE : FD\ and from this point the constructions are similar for both the central conies, the angle AGP being made equal to the angle DKF in each case. Draw now PN perpendicular to the axis, and draw the tangent PT. Then [Props. 48, 49] PN' : CN.NT= CB' : CA' [Prop. 14] and, by simi = FD' .KF. FE, from above ; ar triangles, CN' : PN' = KF' : FD\ .•. CN' : CN.NT= KF' : KF.FE, or ON : NT = KF : FE. And PN : CN = DF : KF. .-.PN:NT=DF.FE. Hence ^^PTN = ^DEF. Proposition 51. [II. 52.] In an ellipse, if the tangent at any point Ρ meet the major axis in T, the angle CPT is not greater than the angle ABA' {where Β is one extremity of the minor a^ns). Taking Ρ in the quadrant AB, join PC. Then PC is either parallel to Β A' or not parallel to it. 78 THE rox/rs OF APOT.LONIUS. First, let PC be parallel to BA'. Then, by parallels, CP bisects ΛΒ. Therefore the β tangent at Ρ is parallel to ΛΒ, and ΔθΡΤ= ΖΛ'ΒΛ. Secondly, suppose that PC is not parallel to Β A', and we have in that case, draAving PN perpendicular to the axis, ZPCN^ ΔΒΛ'ν Δ BAG. whence [Prop. 14] .•. PN' -.CN'^BC' :AC\ PK' : CN' φ PN' : ON. NT. .'. CN^NT. Let FDE be a segment in a circle containing an angle FDF equal to the angle ABA', and let DG be the diameter of the circle bisecting FE at right angles in /. Divide FE in Μ so that EiM : MF = GN : NT, and draw through Μ the chord HK at right angles to EF. From 0, the centre of the circle, draw (JL perpendicular to HK, and join EH, HF. The triangles DFI, BAG are then similar, and FP : ID' = GA' : GB\ Now OD : 01 > LH : LM, since 01 = LM. .•. 01) :Df<LH: Η Μ J TANOENTS, CONJUGATE DIAMETERS AND AXES. 79 and, doublinp^ the antecedents, DG:DI<HK -.HM, whence GI -.IDkEM: MH. But GI ■.ID = FP : TD^ = ΟΑ"" : GB' = GN.NT:PN\ .•. CN. NT : FN' < KM : MH <KM.MH:MH' <EM.MF: MH\ Let ON . NT : PN' = EM. MF : MR\ where R is some point on HK or HK produced. It follows that MR > MH, and R lies on KH produced. Join ER, RF. Now GN . NT : EM . MF = PN^ : RM\ and CN' : ^il/^ = 6'^V . NT : ^il/ . MF (since Ci\r : iVT = EM : J/i?^). .•. CN :EM = PN:RM. Therefore the triangles CPN, ERM are similar. In like manner the triangles PTN, RFM are similar. Therefore the triangles CPT, ERF are similar, and ZCPT= ^ERF; whence it follows that Ζ CPT is less than Ζ EHF, or Ζ ^5^'. Therefore, whether CP is parallel to Β A' or not, the Ζ CPT is not greater than the Ζ ABA'. Proposition 52. (Problem.) [II. 51, 53.] To draw a tangent to any given conic making a given angle iDitli the diameter through the point of contact. I. In the case of the jmrahola the given angle must be an acute angle, and, since any diameter is parallel to the axis, the problem reduces itself to Prop. 50 (1) above. 80 THE COXIOS OF APOLLONIUS. II. In the case of a central conic, the angle CPT must be acute for the Jiyperhula, and for the ellipse it must not be less than a right angle, nor greater than the angle ABA', as proved in Prop. .')!. Suppose θ to be the given angle, and take first the particu- lar case for the ellipse in which the angle θ is equal to the angle ABA'. In this case we have simply, as in Prop. 51, to draw CP parallel to Β A' (or AB) and to draw through Ρ a parallel to the chord A Β (or A'B). Next suppose θ to be any acute angle for the hyperbola, and for the ellipse any obtuse angle less than ABA': and suppose the problem solved, the angle (^PT being e(|ual to Θ. P=^° ^ ^ \ 1 Μ ^ R ο ■^^ < TANGENTS, CONJUGATE DIAMETERS AND AXES. SI Imagine a segment of a circle taken containiug an angle (EOF) equal to the angle Θ. Then, if a point D on the circumference of the segment could be found such that, if DM be the perpendicular on the base EF, the ratio EM .MF : DM^ is equal to the ratio CA"" : CB\ i.e. to the ratio GN .NT : PN\ we should have Ζ CPT = Δθ= Δ EOF, and ON . NT : PN' = EM . MF : ΌΜ\ and it would follow that triangles PCN, PTN are respectively similar to DEM, DFM*. Thus the angle DEM would be equal to the angle PCN. The construction would then be as follows : Draw CP so that the angle PCN is equal to the angle DEM, and draAv the tangent at Ρ meeting the axis ΑΛ' in T. Also let Ρ Ν be pei-pendicular to the axis Λ A'. Then GN . NT : PN' = CA' : GB' = EM. MF : DM\ and the triangles PGN, DEM are similar, whence it follows that the tiiangles PTN, DFM are similar, and therefore also the triangles GPT, EDF*. .•. zCPT= zEDF = ze. It only remains to be proved for the hyperbola that, if the angle PCN be made equal to the angle DEM, CP must necessarily meet the curve, i.e. that the angle DEM is less than half the angle between the asymptotes. If ^ Ζ is per- pendicular to the axis and meets an asymptote in Z, we have EM. MF : DM' = CA' : CB' = GA' : AZ\ .•. EM' : DM' > GA' : AZ\ and the angle DEM is less than the angle ZCA. We have now shown that the construction reduces itself to finding the point D on the segment of the circle, such that EM.MF-.DM'^CA'-.GB'. • These conclusions are taken for granted by ApoUonius, but they are easily proved. H. C. t) 82 THE COXICS OF APOLLONIUS. This is eflfected as follows : Take lengths αβ, /3γ in one straight line such that a/3 : yS7 = CA' : CB\ β^ being measured towards α for the hyperbola and away from α for the ellipse ; and let αγ be bisected in δ. Draw 01 from 0, the centre of the circle, perpendicular to EF\ and on 01 or 01 produced take a point Η such that OH: HI = By: γ/3, (the points 0, H, I occupying positions relative to one another corresponding to the relative positions of δ, γ, β). Draw HD parallel to EF to meet the segment in D. Let DK be the chord through Ό at right angles to EF and meeting it in M. Draw OR bisecting DK at right angles. Then RD : DM =^ OH : HI = 8y : ^β. Therefore, doubling the two antecedents, KD : DM = «7 : 7yS ; so that KM : DM = αβ : β^. Thus KM.MD : DM' = EM.MF : DM' = αβ:β^ = CA' : CB\ Therefore the required point D is found. In the particular case of the hyperbola where CA'= CE^, i.e. for the rectangular hyperbola, we have EM. MF = DM\ or DM is the tangent to the circle at D. Note. ApoUonius proves incidentally that, in the second figure applying to the case of the ellipse, Η falls between / and the middle point (Z) of the segment as follows : Ζ FLI = lz CRT, which is less than ^ Ζ ABA' ; .•. Ζ FLI is less than Ζ ABC, TANGENTS, CONJUOATE DIAMETERS AND AXES. 83 whence CA' : OB" > FP : fiJ >L'l :IL. It follows that αβ : βy > f/ Γ : fL, so that «7 : 7^ > L'L : IL, and, halving the antecedents, δ7 : 7^ > OL : LI, so that Ββ:β^>ΟΙ:ΙΙ. Hence, if Η be such a point that 8β ■.β^ = ΟΙ: IH, I Η is less than IL. 6—2 EXTENSIONS OF PROPOSITIONS 17—19. Proposition 53. [III. 1, 4, 13.] (1) P, Q being any two points on a conic, if the tangent at Ρ and the diameter through Q meet in E, and the tangent at Q and the diameter through Ρ in T, and if the tangents intersect at 0,thm AOPT = AOQE. (2) If Ρ be any point on a hyperbola and Q any point on the conjugate hyperbola, and if T, Ε have the same significance αβ before, then Δ CPE = Δ CQT. (1) Let QV be the ordinate from Q to the diameter through P. Then for the parabola we have TP = PV, [Prop. 12] so that TV=2PV, and CJ EV = AQTV. EXTENSIONS OF PROPOSITIONS 17 — 19. Subtracting the common area OPVQ, AOQE = AOPT. For the central conic we have GV.CT=CP\ 85 or CV :GT=GV':CF'] .•. ACQV:ACQT = ACQV:AGPE; .'. AGQT = AGPE. Hence the sums or differences of the area OTGE and each triangle are equal, or AOPT = AOQE. (2) In the conjur/ate hyperbolas draw GD parallel to the UNIV. 86 THE CO^V/OS OF AFULLUNIUS. tangent at Ρ to meet the conjugate hyperbola in D, and draw QV also parallel to PE meeting CP in V. Then CP, CD are conjugate diameters of both hyperbolas, and QF is drawn ordinate-wise to CP. Therefore [Prop. 15] CV.CT=CP\ or CP:CT=CV:CP = CQ:CE; Λ GP.CE=CQ.Cr. And the angles PCE, QCT are supplementary ; .•. ACQT = ACPE. Proposition 54. [III. 2, 6.] // we keep the notation of the last proposition, and if R he EXTENSIONS OF PROPOSITIONS 17 — 10. 87 any other point on the conic, let RU be drawn parallel to QT to meet the diameter through Ρ in U, and let a parallel throu(/h R to the tangent at Ρ meet QT and the diameters through Q, Ρ in H, F, W respectively. Then A HQF = quadrilateral HTUR. Let RU meet the diameter through Q in M. Then, as in Props. 22, 23, Ave have Δ RMF= quadrilateral QTUM ; .•., adding (or subtracting) the area HM, Δ HQF= quadrilateral HTUR. Proposition 55. [III. 3, 7, 9, 10.] //' we keep the same notation as in the last proposition and take two points R', R on the curve luith points H' , F', etc. corre- sponding to H, F, etc. and if, further, RU, R'W intersect in I and R'U', RW in J, then the quadnlaterals F'IRF, lUU'R' are equal, as also the quadrilaterals FJR'F', JU'UR. [N.B. It will be seen that in some R cases (according to the positions of R, R') the quadrilaterals take a form like that in the margin, in which case F'IRF must be taken as meaning the diflfereuce between the triangles F'MI, RMF.] I. We have in figs. 1, 2, 3 Δ HFQ = quadrilateral HTUR, [Prop. .54] AH'F'Q = quadrilateral H'TU'R', .•. F'H'HF=H'TU'R'~HTUR = IUU'R' + (IH); whence, adding or subtracting IH, F'IRF = IUU'R' (1). 88 THE CONIL'S OF APOLLONIUS. and, adding {IJ) to bulh, FJR'F'=JU'UR. Fig. 1. II. In Hws. 4, 5, G we have [Prop.s. IS. 53] so that Δ GQT = quadrilateral CU'R'F', EXTENSIONS OF PROPOSITIONS 17—19. and, adding the quadrilateral CF'H'T, we have AH'F'Q = quadrilateral H'TU'R'. Fig. 5. Similarly Δ HFQ = HTUR; and we deduce, as before, F'lRF^IUU'R Thus e.g. in fig. 4, AH'F'Q" - AHFQ = H'TU'R- HTUR ; .•. F'H'HF={R'H)-{RU'), and, subtracting each from {IH), F'lRF^IUU'R'. In fig. 6, F'H'HF = H'TU'R' - AHTW+ ARUW, .(1). Fig. (>. 90 THE COXICS UF AFOLLONIUS. and, adding (///) to each side, F'IRF = H'TU'R' + H'TUI = IUU'R' (1). Then, subtracting (//) from each side in fig. 4, and sub- tracting each side from (IJ) in figs. 5, 6, we obtain FJR'F' =JU'UR (2), (the quadrilaterals in fig. 6 being the differences between the triangles FJM', F'R'M' and between the triangles JU'W,RUW respectively). III. The same properties are proved in exactly the same manner in the case where P, Q are on opposite branches, and the quadrilaterals take the same form as in fig. 6 above. Cor. In the particular case of this proposition where R' coincides with Ρ the results reduce to EIRF=APUI, PJRU = PJFE. Proposition 56. [III. 8.] //' PP', QQ' be two diameters and the tangents at P, P', Q, Q' be drawn, the former two meeting QQ' in E, E' and the latter two meeting PP' in T, T', and if the parallel through P' to the tangent at Q meets the tangent at Ρ in Κ luhile the parallel through Q' to the tangent at Ρ meets the tangent at Q in K', then the quadrilaterals (EP'), (TQ') are equal, as also the quadri- laterals (E'K), {T'K'). Since the triangles CQT, CPE are equal [Prop. 53] and have a common vertical angle, CQ.CT=CP.CE; .•. CQ '. CE = GP : GT, EXTENSIONS OF PROPOSITIONS 17 — 19. 91 whence QQ' : EQ = PP' : TP, and the same proportion i.s true for the squares ; .•. AQQ'K' : AQEO = APP'K : ΑΡΤΟ. And the consequents are equal ; .•. AQQ'K' = APP'K, and, subtracting the equal triangles CQT, CPE, we obtain (EP') = (TQ') (1). Adding the equal triangles CP'E', CQ'T' respectively, we have {E'K) = {T'K') (2). Proposition 57. [III. -r>, 11, 12, 14.] (Application to the case where the ordinates through R, R, the points used in the last two propositions, are drawn to a secondary diameter.) (I) Let Gv be the secondary diameter to which the ordi- nates are to be drawn. Let the tangent at Q meet it in t, and let the ordinate Rw meet Qt in h and CQ in /'. Also let Ri, parallel to Qt, meet Cv in a. Then [Prop. 19] ARm- ACfw= ACQt (A) 92 THE VOXJCS OF APOLLONIUS. and, subtracting the (iiuidnlateral GiuhQ, ARuw ~A}tQf= Ahtiu ; .•. AhQf= C[na.an\siteYa\ htuR. (2) Let R'lu' be another ordinate, and h', w' &c. points corresponding to h, ιυ, &c. Also let Ru, R'lu meet in i and Riu, R'u m j. Then, from above, Ah'Qf = }itiifR', and AhQf = htuR. Therefore, subtracting, f'h'hf = iuii'R — (hi) and, adding (hi), fiRf=mu'R' (1). If we add {i}) to each, we have fjR'f=ju'uR (2). [This is obviously the case where Ρ is on the conjugate hyperbola, and we deduce from (A) above, by adding the area CwRM to each of the triangles Ruw, Gfw, ACuM'- ARfM= ACQt, a property of which ApoUonius gives a separate proof.] EXTENSIONS OF PROPOSITIONS 17 — 19. 93 Proposition 58. [III. 15.] In the case where P, Q are on the oHginal hijperhola and R on the conjugate hyperbola, the same properties as those formu- lated in Propositions 55, 57 still hold, viz. ARMF^ ACMU= ACQT, and F'IRF=IUU'R'. Let D'D" be the diameter of the conjugate hyperbola parallel to R U, and let QT be drawn ; and from D' draw DG parallel to PE to meet CQ in G. Then D'D" is the diameter conjugate to GQ. Let ρ be the parameter in the conjugate hyperbola corre- sponding to the transverse diameter D'D", and let ρ be the parameter corresponding to the transverse diameter QQ' in the original hyperbola, so that I . CQ = CD", and ζ . CD' = CC^. ΝοΛν we have [Prop, 23] Oq:QE = p:2QT = ^^:QT: 94 THE COXTCS OF APOLLONIUS. .. D'C:CG = ^:QT = ^.CQ:CQ.QT = CD":GQ.QT. Hence DV.CG=CQ.QT, or AD'CG= AOQT (1). Again. CM.MU=CQ.QT = (CQ: !).(/; :2ρΓ) = (p'.D'D").{OQ.QE) = (p : D'D") .(R3I: MF) (2). Therefore the triangles GMU, RMF, D'CG, being respec- tively half of equiangular parallelograms on CM (or Rv), RM (or Cv), CD', the last two of which are similar while the sides of the first two are connected by the relation (2), have the property of Prop. 16. .•. ARMF- ACMU= AD'CG= ACQT (3). If R' be another point on the conjugate hyperbola, we have, by subtraction, R'JFF - RMM'J = MUU'M', or RJFF = RUU'J. And, adding (IJ), F'IRF=IUU'R' (4) RECTANGLES UNDER SEGMENTS OF INTERSECTING CHORDS. Proposition 59. [III. 16, 17, 18, 19, 20, 21, 22, 23.] Case I. If OP, OQ be two tangents to any conic and Rr, R'r two chords parallel to them respectively and intersecting in J, an internal or e.dernal point, then OP': OQ' = RJ.Jr:R'J.Jr: (a) Let the construction and figures be the same as in Prop. 55. We have then RJ.Jr = RW'^JW\ and RW':JW'=ARUW: AJU'W; .•. RW'~JW':RW' = JU'rR: ARUW. But R W : 0P'= AR UW : Δ OPT ; .•. RJ.Jr : OP' = JU'UR: AOPT (1). Again R'J . Jr = R'M" ~ JM" and R'M" : JM" = AR'F'M' : AJFM', m R'M" ~ JM" : R'M" = FJR'F' •. A R'F'M'. But R'M" :0Q'= A R'F'M' : A OQE ; .•. R'J.Jr': OQ' = FJR'F: AOQE (2). 96 THE COXICS OF APOLLONIUS. Comparing (1) and (2), we have JU'UR = FJR'F, by Prop. 55, and Δ OPT = Δ OQE, by Prop. 53. Thus BJ. Jr : OP' = R'J. Jr' : 0Q\ or OP' : OQ' = RJ. Jr : R'J. Jr'. (b) If we had taken the chords R'r^', Rr^ parallel respec- tively to OP, OQ and intersecting in /, an internal or external point, we should have established in the same manner that Or-:OQ' = R'I.Ir;:RI.h\. Hence the proposition is completely demonstrated. [Cor. If /, or J, which may be any internal or external point be assumed (as a particular case) to be the centre, we have the proposition that the rectangles under the segments of intersecting chords in fixed directions are as the squares of the parallel semi-diameters.] Case II. If Ρ be a point on the conjugate hyperbola and the tangent at Q meet GP in t ; if further qq' be draivn through t parallel to the tangent at P, and Rr, R'r' be tiuo chords parallel respectively to the tangents at Q, P, and intersecting at i, then tQ' : tq" = Ri . ir : R'i . ir'. Using the figure of Prop. 57, we have Ri.ir = Mi''-MR\ and Mi^ : MR' = AMfi : AMfR. Hence Ri . ir : MR' = fiRf : Δ MfR. Therefore, if QC, qq' (both produced) meet in L, Ri.ir:tQ'=fiRf: AQtL (1). Similarly, R'i . ir' : R'w" = iuu'R' .: Δ R'u'w' : .• . R'i . ir' : tq' = iuu'R' : AtqK (2), where qK is parallel to Qt and meets Ct produced in K. RECTANGLES UNDER SEGMENTS OF INTERSECTING CHORDS. 97 But, comparing (1) and (2), we have f'iRf= iuu'R, and Δ tqK = Δ CLt + ACQt= A QtL. .•. Ri.ir:tQ' = Ii'i.ir':tq\ or tQ':tq^ = Ri.ir:R'i.ir'. [Prop. 57] [Prop. 19] Case III. If PP' he a diameter and Rr, R'r' he cJwrds parallel respectively to the tangent at Ρ and the diameter PP' and intersecting in I, then RI.Ir:R'I.Ir' = p:PP'. If RW, R'W are ordinates to PF, ρ : PP' = RW : CW - CP' = R'W":CW"~CP' = RW'-'R'W"':CW' = RI.Ir.R'I.Ir'. [Prop. 8] CW Case IV. If OP, OQ he tangents to a hyperhola and Rr, R'r' he two chords of the conjugate hyperhola parallel η to OQ, OP, and meeting in I, then OQ':OP' = RI.Ir.R'I.Ir'. Using the figure of Prop. 58, we have OQ' : Δ OQE = RiW : Δ RMF = MP: AM IF' = RI.Ir: ARMF- AMIF' ^ Ri.Ir: F'lRF, H. c. 7 98 THE COXirs OF APOLLONIUS. and, in the same way, OF': A()PT=R'r.Ir': AR'U'W - AIUW = R'I.Ir':IUU'R'; whence, by Props. 53 and 58, as before, ()Q':RI.Iv=OP'.R'I.Ir', or Oqt: OP' = RI.Ir.R'I.Ir'. Proposition 60. [III. 24, 25, 26.] If Rr, R'r' he chords of conjugate hi/perbolas meeting in and parallel respectively to conjugate diameters PP', DD', then R0.0r+^^,.RO.0r' = 2CP' Γ RO.Or R'O.Or' „1 Let Rr, R'r meet the asymptotes in K, k ; K', k', and CD, CP in w, W respectively. Draw LPL', the tangent at P, meeting the asymptotes in L, L', so that PL = PL'. Then LP.PL'=CD\ and LJ' . PL' : GP' = CD^ : CP\ Now LP : CP = K'O : OK, PL':CP = 0k':0k; .•. CV : CP' = K'O . Ok' : KO . 0/.•. RECTANGLES ΓΝΠΚΙΙ SEGMENTS OF INTERSECTING CHORDS. 09 [From this point Apollonius distinguishes five cases: (1) where is in the angle LCL', (2) where is on one of the asymptotes, (8) where is in the angle LCk or its opposite, (4) where is within one of the branches of the original hyperbola, (5) where lies within one of the branches of the conjugate hyperbola. The proof is similar in all these cases, and it will be sufficient to take case (1), that represented in the accom- panying figure.] We have therefore CD' : CP' = K'O . Ok' + C'D' : KO . Ok + CP' = K'O . Ok' + K'R . R'k' :KO.Ok + CP' = K' W" -0W"-\- R W" - K' W" : Ow^ - Kiu' + CP' = R W" - W" : Riv' - Kw' - Riv' + Οιυ' + CP' = RO . Or' : RK . Kr + GP' - RO . Or = RO . Or' : 2CP' - RO . Or (since Kr = Rk), fip2 whence RO .Or + ^,.R'0. Or' = 2 CP\ RO.Or RO . Or ' or ^p2 + ^^, [The following proof serves for all the cases : we have RW - CD' : CW" = CD' : CP" and Cid" : Riu'' - CP' = CD" : CP' ; ... R'W" - Cid" - CD' : CF' - (Rtu' - CW") = CD' : CP\ so that + RO . Or' - CD' : CP' ±RO.Or= CD' : CP', whence ± RO . Or' : 2CP' ± RO . Or = CD' : CP' RO.Or' RO.Or „, —CD^-^-CP^-^-^ 7—2 100 THE COXICS OF APOLLONIUS Proposition 61. [III. 27, 28, 29.] If in ai} ellipse or in conjugate hyj^erholas two chords Rr, R'r he drawn meeting in and parallel respectively to two conjugate diameters FP', DD', then (1) for the ellipse RO' + Or' +^^3 {RV + Or") = 4CP^ RO^+Or' RO'+Oj''\ or ^p, + ^^, -4, and for the hyperbolas RO' + Or' : R'O' + Or" = CP' : CD\ Also, (2) if R'r' in the hyperbolas meet the asymptotes in K', k', then K'O' + Ok" + ^GD' : RO' + Or' = CD' : CP\ (1) We have for both curves CP':CD' = PW.WP'.RW' = R'w": Div'.w'B' = CP' + Ρ W . WP' ± R'w" : CD' + R W + Dw' . w'D' INTERSECTING CHORDS. 101 (taking the upper sign for the hyperbolas and the lower for the ellipse) ; .•. CP' : CD' = CP' ± CW" + Ρ W. WP : CD' + Cw' ± Dw'.w'D', whence, for the hyperbolas, CP : CD' = CW" + GW^ : Cw' + Cw" = UR0' + 0r'):^{RO' + 0r"), or RO' + Or' : RV + Or" = CP' : CD' (A), while, for the ellipse, CP' : CD' = 2CP' -(CW" + CW) : Cw" + Cw^ = ^CP' - {RO' + Or') : {RV + Or"), , RO'+Or'_^ R'O' + Or" whence — ^pi — Η jjjji =4 {B). (2) We have to prove that, in the hyperbolas, R'O' + Or" = K'O' + Ok" + 2CD'. Now R'O' - K'O' = R'K" + 2R'K' . K'O, and Or" - Ok" = r'k" + 2r'k' . k'O = R'K" + 2R'K'.kO. Therefore, by addition, R'O^ + Or'-^ _ K'O' - Ok" = -IR'K' {R'K' + K'O + Ok') = 2R'K'.R'k' = 2CD'. ... R'O' + Or" = K'O' + Ok" + 2CD\ whence K'O' + Ok" + 2CD' : RO' + Or' = CD' : CP', by means of (A) above. HARMONIC PROPEllTIES (JF POLES AND POLARS. Proposition 62. [III. :}(), 31, 82, :v.i U.] TQ, T(j being taiKjents to a Injperhula, if V he the middle point of Qq, and if TM he drawn parallel to an asymptote meeting the curve in. R and Qq in M, luhile VN parallel to an asymptote meets the curve in R' and the parallel through Τ to tlie chord of contact in N, then TR = RM, VR' = R'N*. I. Let CV meet the curve in P, and draw the tangent PL, which is theretbrc parallel to Qq. Also draAv the ordinates RW, R'W to CP. Then, since the triangles CPL, 'TWR are similar, R W : TW = PL' : CP' = CD' : CP' = RW':PW. WP'; .•. TW'' = PW. WP'. • It will be observed from this proposition and the next that Apollonius begins with two particular cases of the general property in Prop. 64, namely (<i) the caHc where the transversal is parallel to an asymptote, (l>) the case where the chord of contact is parallel to an asymptote, i.e. where one of the tangents IB an aHymptute, or a tangent at infinity. HARMONK! PROPERTIES OF POLKS AND POLARS. 103 Also CV.CT=CP', .•. PW. WF' + GP'=GV.CT+TW\ or CW' = CV.CT+TW\ whence CT(CW+TW) = CV. CT, and TW= WV. It follows by parallels that TR = RM (1 ). Again GP' : PU = W V : W'R" ; .•. W'V: W'R"' = PW' . WP' : W'R'\ so that PW'.W'P'= W'V\ And GV.CT=GP'; .•. 6ΊΓ^ = CF.Cr+lΓF^ whence, as before, TW = WV, and NR' = R'V (2). II. Next let Q, q be on opposite branches, and let P'P be the diameter parallel to Qq. Draw the tangent PL, and the ordinates from R, R', as before. Let TM, GP intersect in K. Then, since the triangles GPL, KWR are similar, GP' : PU = KW : WK\ and GP' : GD' = PW . WP' : WR' ; .•. KW' = PW.WP'. Hence, adding GP\ GW'[=Rw''] = KW' + GP\ But Rw' : ii W' + CT^ = 2^i<;•^ : R W' + PZ^ by similar triangles. Therefore Tw' = RW' Λ- GD' = Gw' + GV.GT, 104 THE COXICS OF APOLLONIUS. whence Tw — Cw = CV, or Ί\υ = wV\ .•. TR = RM (1). Again rP' : ΡΓ = Ρ W . W'P' : R' W" = PW'. W'P' + CP": R' W" + CD" = CW":Ow''^-CV.GT. Also GP" : PU = R'w'^ :w'V'\ .•. w'V = Cw" + cv.cr, wliciicc, as before, Tw' = w'F, and, by parallels, NR' = R'V. (2). III. The particular case in which one of the tangents is a tangent at infinity, or an asymptote, is separately proved as follows. Let LPL' be the tangent at P. Draw PD, LM parallel to CL\ and let LM meet the curve in R and the straight line Pi^ drawn through Ρ parallel to CL in M. Also draw RE parallel to CL. Now LP = PL'; .•. PD - CF = FL', FP = CD = DL. And FP.PD = ER. RL. [Prop. 34] But ER = LC = 2CD = 2FP: .•. PD = 2LR, or LR = RM. Proposition 63. [III. 35, 36.] // PL, the tangent to a hyperbola at P, meet the asymptote in L, and if PO be parallel to that asymptote, and any straight line LQOQ' be drawn meeting the hyperbola in Q, Q' and PO in U, then Ur : LQ = QV : OQ. HARMONIC PROPERTIES OF POLES AND POLARS. 105 Wc have, drawing parallels through L, Q, P, Q' to both asymptotes as in the figures, LQ = Q'L' : whence, by similar triangles, DL = IQ' = CF .•. CD = FL, and CD.DL = FL: LD = Q'L : LQ = MD : DQ. Hence {HD) : (Ζ>ΤΓ) = (i/C) : {CQ) = {MC):{EW), since (CQ) = {CP) = {E \V). [Prop. 34] Therefore {MG) : {EW) = {MC) ± (HD) : (EW) ± (DW) =^{MH):{EU) (1). Now {DG) = (HE). [Prop. 34] Therefore, subtracting CX from both, {BX) = {XH), and, adding (XU) to each, (EU) = (HQ). Hence, from (1), since (EW) = (CQ), (MG):(CQ) = (MH):(HQ), or LQ' : LQ = Q'O : OQ. [Apollonius gives separate proofs of the above for the two cases in which Q, Q' are (1) on the same branch, and (2) on opposite branches, but the second proof is omitted for the sake of brevity. Eutocius gives two simpler proofs, of which the following is one. Join PQ and produce it both ways to meet the asymptotes in R, R. Draw PV parallel to CR' meeting QQ' in V. 106 TUE CVXKJS OF Al'OLLOMUS. Then LV=VL'. But ρ/. = (//.': .•. QV= VQ'. N( QV: VL' = QP.PR' = PQ:QR = OQ : QL. 2QV : 2VL' = OQ : QL, QQ' : OQ = LL' : QL ; .•. QO:OQ=LQ':LQ.-\ Proposition 64. [III. 37, 38, 39, 40.] (1) If TQ, Tq he tangents to a conic and any straight line he drawn through Τ meeting the conic and the chord of contact, the straight line is divided harmonically ; (2) // any straight line he drawn through V, the middle point of Qq, to meet the conic and the parallel through Τ to Qq [or the polar of the point F], this straight line is also divided harmonically ; i.e. in the figures drawn below (1) RT:TR' = RI:IR', (2) RO:OR' = RV: VR'. IIAUMUNIC I'ROFERTIES OF POLES AND I'OLAllS. 107 Let TF be the diameter bisecting Qq in V. Draw as usual IIRFW, H'R'F'W, EF ordinate-wise to the diameter TF; and draw RU, R'U' parallel to QT meeting TF in U, U'. (1) We have then R'r:IR' = H'Q':HQ' = AH'F'Q: AHFQ = H'TU'R' : HTUR. [Props. 54, 55] Also RT : TR' = R' U" : R U' = AR'U'W: ARUW; and at the same time RT : TR' = TW" : TW . = ATH'W: ATHW] .•. Rr:TR= AR'U']V' ~ ATH'W: ARUW - ATHW = H'TU'R' : HTUR = RT : IR\ from above. .•. RT : TR' = RI : IR'. (2) We have in this case (it is unnecessary to give more than two figures) RV: VR" = RU':R'U" = ARUW: AR'U'W. 108 THE CUSICS OF AIOLLOXIUS. Also MV: VR" = HQ':QH" = AHFQ : AH'F'Q = HTUR : H'TU'R. .•. RV: VR" = HTUR + ARUW : H'TU'R' ± AR'U'W = ATHW: ATH'W = TW':TW'* = RO':OR"; that is. RO : OR' = RV : VR'. INTERCEPTS MADE ON TWO TANGENTS BY A THIRD. Proposition 65. [III. 41.] If the tangents to a 'parabola at three points P, Q, R form a triangle pqr, all three tangents are divided in the same propor- ti&n, or Pr : rq = rQ : Qp = qp : pR Let V be the middle point of PR, and join qV, which is therefore a diameter. Draw T'TQW parallel to it through Q, meeting Pq in Τ and qR in T. Then QW is also a diameter. Draw the ordinates to it from P, R, viz. PU, RW, which are therefore parallel to pQr. 110 THE COXICS OF APOLLONIUS. Now, if ^F passes through Q, the proposition is obvious, and the ratios will all be ratios of equality. If not, we have, by the properties of tangents, drawing EBF the tangent at the point Β where qV meets the curve, TQ = QU, T'Q=QW, qB = BV, whence, by parallels, Pr = rT, Tp=pR, qF=FR. Then (1) rP.PT=EP:Pq=l: 2, and, alternately, rP : PE = TP : Pq = OP : PV, Avhence, doubling the consequents, rP :Pq=OP: PR, and Pr:rq = PO:OR (1). (2) rQ'.Qp = PU:RW, since PU=2rQ, and RW = 2pQ ; Qp = PO: OR (2). Rq=pR:RT', Rp^qR: RT = VR : RO. Therefore, doubling the antecedents, qR:Rp = PR: RO, whence qp : pR = PO : OR (3). It follows from (1), (2) and (3) that Pr : rq = vQ : Qp = qp : pR. (3) FR and, alternately, FR INTERCEPTS MADE ON TWO TANGENTS BY A THIRD. Ill Proposition 66. [III. 42.] If the tangents at the eairemities of a diameter PP' of a central conic he drawn, and any other tangent meet them in r, r respectively, then Pr.P'r' = GD\ Draw the ordinates QV, Qv to the conjugate diameters PP' and DD' ; and let the tangent at Q meet the diameters in T, t respectively. If now, in the case of an ellipse or circle, CD pass through Q, the proposition is evident, since in that case rP, CD, r'P' will all be equal. If not, we have for all three curves CT.GV=CP\ so that CT:CP = CP: CV = CT-CP:CP -^GV = PT:PV: .•. CT: GP' = PT :PV, whence GT:P'T = PT: VT. Hence, by parallels, Gt : P'r' = Pr : QV = Pr:Gv; .•. Pr.P'r'^Gv.Gt = GD\ 112 THE COyiCS OF APOLLONIUS. Proposition 67. [III. 43.] If a tangent to a Jii/perbola, LPL', meet the asymptotes in L, L', the triangle LCL has a constant area, or the rectangle LC . CU is constant. Draw PD, PF parcallel to the asymptotes (as in the third figure of Prop. 62). ΝοΛν LP = PL'; .•. CL = 2CD = 2PF, CL' = 2CF=2PD. .•. LG.CL' = ^DP.PF, which is constant for all positions of P. [Prop. 34] Proposition 68. [III. 44.] If the tangents at P, Q to a hi/perhola meet the asymptotes respectively in L, L' ; M, M', then LM', L'M are each parallel to PQ, the chord of contact. Let the tangents meet at 0. We have then [Prop. 67] LC.CL' = MC.CM', so that LC\ CM' = MC: CL'\ .•. LM' , L'M arc parallel. It follows that OL : LL' = OM' : M'M, or, halving the consequents, OL: LP=OM':M'Q; .•. l.M', J'Q aru parallfl. FOCAL PROPERTIES OF CENTRAL CONICS. The foci are not spoken of by Apollonius under any equiva- lent of that name, but they are determined as the two points on the axis of a central conic (lying in the case of the ellipse between the vertices, and in the case of the hyperbola within each branch, or on the axis produced) such that the rectangles AS.SA', AS' .S'A' are each equal to "one-fourth part of the figure of the conic," i.e. \p„.AA' or CB"^. The shortened expression by which S, S' are denoted is τα βκ τή<; τταραβοΧής <γινόμ€να σημεία, " the points arising out of the application." The meaning of this Λνϋΐ appear from the fiill description of the method by which they are arrived at, which is as follows : iav τω τ€τάρτω μέρει τον εΓδους• "σον τταρα τον άξονα τταραβΧηθτ} 60' €κάτ€ρα iirl μεν της υττερβοΧής καΐ των αντικειμένων νττερβάΧλον e'iBei τετραηώνω, iirl 8e τή^ εΧΧείψεως εΧΧεΐττον, " if there be applied along the axis in each direction [a rect- angle] equal to one-fourth part of the figure, in the case of the hyperbola and opposite branches exceeding, and in the case of the ellipse falling short, by a square figure." This determines two points, which are accordingly τα εκ τΓ/ς 7ΓαραβοΧ7}<^ ηενηθέντα Η. C. 114 THE COXJCS OF APOLLONIUS. σημύα. That is, we are to suppose a rectangle applied to the axis as base which is equal to CB^ but which exceeds or falls short of the rectangle of equal altitude described on the ivhole axis by a square. Thus in the figures drawn the rectangles AF, ^'/'are respectively to be equal to CB\ the base AS' falling short of AA' in the ellipse, and the base A'S exceeding A'A in the hyperbola, while S'F or SF is equal to S'A' or SA respectively. The focus of a parabola is not used or mentioned by Apollonius. Proposition 69. [III. 45, 46.] If Ar, A'r' , the tangents at the extremities of the axis of a central conic, meet the tangent at any point Ρ in r, r' respectively, then (1) 7ύ' subtends a right angle at each focus, S, S' ; (2) the angles rr'S, A'r'S' are equal, as also are the angles r'rS', ArS. (1) Since [Prop. 60] rA.A'r' =^Cn' = AS .SA', by definition, rA :AS=SA' : A'r'. FOCAL PROPERTIES OF CENTRAL CONICS. 115 Hence the triangles rAS, SAY are similar, and zArS= zA'Sr'; .•. the angles iSA, A'Sr' are together equal to a right angle, so that the angle rS?-' is a right angle. And similarly the angle rSV is a right angle, (2) Since rSr', rS'r' are right angles, the circle on rr' as diameter passes through S, S' ; .•. Ζ rr'S = Ζ rS'S, in the same segment, = Ζ S'r'A', by similar triangles. In like manner Ζ r'rS' = Ζ AiS. Proposition 70. [III. 47.] If, in the same ficjures, be the intersection of rS', r'S, then OP loill he perpendicular to the tangent at P. Suppose that OR is the perpendicular from to the tangent at P. We shall show that Ρ must coincide with P. For Ζ Or'R = ζ S'r'A', and the angles at R, A' arc right ; .*. the triangles Or'R, S'r'A' are similar. 8—2 116 THE coyics ov apollonius. Thereioie A'r' : r'R = S'r' : r'O = Sr : I'O, by similar triangles, = Ar : rR, because the triangles ArS, RrO are similar; .•. r'R : Rr = A'r' : Ar = A'T : TA (1). Again, if PN be drawn perpendicular to the axis, we have [Prop. 13] A'T -TA^A'N : Ν A = r'P : Pr, by parallels. Hence, from (1), r'R : Rr = r'P : Pr, and therefore R coincides with P. It follows that OP is perpendicular to the tangent at P. Proposition 71. [III. 48.] The focal distances of Ρ make equal angles with the tangent at that point. In the above figures, since the angles rSO, OPr are right [Props. 69, 70] the points 0, P, r, S are concyclic ; .•. Ζ SPr = ζ SOr, in the same segment. In like manner Ζ S'Pr' = Ζ S'Or', and the angles SOr, S'Or' are equal, being the same or opposite angles. Therefore Ζ SPr = Ζ S'Pr'. Proposition 72. [III. 49, 50.] (1) If, from either focus, as S, SY be drawn perpendicular to the tangent at any point P, the angle AY A' will be a right angle, or the locus of Υ is a circle on the aris A A' as diameter. (2) The line drawn through C parallel to either of the focal distances of Ρ to meet the tangent ivill be equal in length to CA, or CA'. FOCAL PROPERTIES OF CENTRAL CONICS. Il7 Draw iSiF perpendicular to the tangent, and join ΛΥ, VA'. Let the rest of the construction be as in the foregoing proposi- tions. We have then (1) the angles rAS, rYS are right ; .. A, r, Y, S are concyclic, and ZAYS=ZArS = Ζ 7''8A', since Ζ rSi^' is right = Ζ 1^'YA', in the same segment, S, Y, r', A' being concyclic ; .". , adding the angle SYA', or subtracting each angle from it, Ζ A Υ A' = Ζ SYr' = a right angle. Therefore Υ lies on the circle having A A' for diameter. Similarly for F'. (2) Draw GZ parallel to SP meeting the tangent in Z, and draw S'K also parallel to SP, meeting the tangent in K. Now AS.SA' = AS\S'A', whence AS = S'A', and therefore CS = CS'. Therefore, by parallels, PZ=ZK. Again Ζ S'KP = Ζ SP F, since SP, S'K are parallel, = ^S'PK; [Prop. 71] .•. S'P = S'K. And PZ = ZK; .•. S'Z is at right angles to the tangent, or Ζ coincides with F'. But F' is on the circle having A A' for diameter ; .•. GT = CA, or CA'. And similarly for GY. 118 THE COXICS OF APOLLONIUS. Proposition 73. [III. 51, 52.] In an ellipse the sum, and in a hyperbola the difference, of the focal distances of any point is equal to the a.xis Λ A'. We have, as in the last proposition, if SP, CY', S'K are parallel, S'K = ST. Let S'P, CY' meet in M. Then, since SG = GS', SP = 2GM, S'P = S'K=2MY': .•. SP + S'P = 2(CM + MT) = 2GY' = AA'. [Prop. 72] THE LOCUS WITH RESPECT TO THREE LINES &c. Proposition 74. [Ill 53.] If PP' he a diameter of a central conic, and Q any other point on it, and if PQ, P'Q respectively meet the tangents at P', Ρ in R, R, then PR.P'R = DD'\ 120 THE ayxics of apollonius. Draw the ordinate QF to ΡΓ. Now ρ : PP' = Q V -.PV.P'V [Prop, i = (QV:PV).(QV:P'V) = (PR : PP') . (PR : PP'), by similar triangles Hence ρ : PF = PR .P'R . PP'\ Therefore PR . PR = ρ . PP' = DD'\ Proposition 75. [III. .34, .-)6.] TQ, TQ' beinij tiuo tangents tu a conic, and R any other point on it, if Qr, Q'r' he draimi parallel respectively to TQ', TQ, and if Qr, Q'R meet in r and Q'r', QR in )•' , then Qr . Q'r' : QQ"' = (PV' : ΡΓ) χ {TQ . TQ' : QV\ where Ρ is the point of contact of a tangent parallel to QQ'. THE LOCUS WITH RESPECT TO THREE LINES ETC. 121 Draw through R the ordinate β ΤΓ (parallel to QQf) meeting the curve again in R and moi-ting TQ, TQ' in K, K' respec- tively ; also let the tangent at Ρ meet TQ, TQ in L, L'. Then, since PV bisects QQ', it bisects LL , KK\ RK also. Now QU : LP. PL' = QL• : LP' = QK':RK.KR' [Prop. 59] = QK':RK.RK'. But QL . Q'L' : QL' = QK . Q'K' : QK\ Therefore, ea: aequali, QL . Q'L' : LP . PL' = QK . Q'K' : RK . RK' = (Q'K':K'R).(QK:KR) = {Qr:QQ').{Q'r' -.QQ') = Qr.Q'r':QQ''- Qr . Q'r' : QQ" = QL . Q'L' : LP . PL' = {QL . Q'L' : LT. TL) . {LT . TL' : LP . PL) = {PV':Pr).CTQ.TQ':QV'). 122 THE t'OXKM <JF APOLLONIUS. Proposition 76. Llll. :.x] If the tangents are tangents to opposite branches and meet in t, and if tq is half the chord through t parallel to QQ', while R, r, r have the same meaning as before, then Qr.Q'r':QQ"=tQ.tQ':tq\ Let RM be the chord parallel to QQ' drawn through R, and let it meet tQ, tQ' in L, L'. Then QQ', RR', LL' are all bisected by tv. Now [Prop. 59] tq'.tQ'=R'L.LR:LQ' = L'R.RL:LQ\ But tQ' : tQ . tQ' = LQ' : LQ . L'Q'. Therefore, ex aequali, tq' : tQ . tQ' = L'R .RL-.LQ. L'Q' = (L'R : L'Q') . (RL : LQ) = {QQ':Qr).(QQ':Q'r') = QQ":Qr Thus Qr.Q'r':QQ"=tQ.tQ':tq\ [It is easy to sec that the last two propositions give the property of the three-line locus. For, since the two tangents and the chord of contact are fixed while the position of R alone varies, the result may be expressed thus, Qr . Q'r = (const.). THE LOCUS WITH RESPECT TO THREE LIXES ETC. 123 Now suppose Q,, Q,, Γ, in the accompanying figure substi- tuted for Q, Q', Τ respectively in the first figure of Prop. 75, and we have Q^r . Qy = (const.) Draw Rq^, Rq.^ panillel respectively to T,Q,, T^Q^ and meeting Q^Q^ in q^, q^. Also let Rv^ be drawn parallel to the diameter CT, and meeting QJ^^ in v,. Then, by similar triangles, Q,r:Rq; = ClQ.--Q.q:, Qy:Rq,= Q,Q,:Q,q, Hence Q,r . Q/ : % . Rq,' = Q,Q,' : Q,q, . Q,q,. But Rq^ . %/ : Rv^' = T,Q, . T^Q, : Γ, V\ by similar triangles .•. Rq^ . Rq^ : jRy/ = (const.). Also QiQ^ is constant, and Q{i' . Q^/"' is constant, as proved. It follows that -flv," : Qii, . Q/y/ = (const.). But Rv^ is the distance of R from Q,^.^, the chord of contact measured in a fixed direction (parallel to 0T^)\ and Qj^,, QjQ'j' are equal to the distances of R from the tangents jTjQj, jTjQj respectively, measured in a fixed direction (parallel to the chord of contact). If the distances arc measured in any 124 THE CO.yiCS OF Al'OLLOXlUS. other fixed directions, they will be similarly related, and the constant value of the ratio will alone be changed. Hence R is such a point that, if three straight lines be drawn from it to meet three fixed straight lines at given angles, the rectangle contained by tw^o of the straight lines so drawn bears η constant ratio to the square on the third. In other words, a conic is a "three-line locus" where the three lines are any two tangents and the chord of contact. The four-line locus can be easily deduced from the three- line locus, as presented by Apollonius, in the following manner. If QiQjQgQ^ be an inscribed quadrilateral, and the tangents at Q^, Q„ meet at Γ,, the tangents at Q^, Q^ at jT^ and so on, suppose Bq^, Rq^ drawn parallel to the tangents at Q^, Q^ respectively and meeting Q^Q^ in q^, q^ (in the same way as Rq^ , Rq•^ were drawn parallel to the tangents at Q, , Q^ to meet QxQi)' ^i^d let similar pairs of lines Rq^, Rq^' and Rq^, Rq^ be drawn to meet Q,Q^ and Q^Q, respectively. Also suppose Rv^ drawn parallel to the diameter GT^, meet- ing Q,Qj in I'j, and so on. Then we have Q^^U ■ Qs^s = ^'2 • ^V 1 ^vhere k^, k^, k„ k, are Qsqs'Q.q: = K-R<[ constants. Hence we derive Rv^'-Rv: 'Q.qrQs9:'Q.q/Q.q. where k is some constant. But the triangles Qtq^qt', Q^qsi ♦^^.c. are given in species, SIS all their sides are in fixed directions. Hence all the ratios ^'■'', etc. are constant; Rv^.Rv, ^ ^, THE LOCUS WITH RESPECT TO THREE LINES ETC. 12; But Bv^, Rv^, Rv^, Ri\ are straight lines drawn in fixed directions (parallel to CT,, etc.) to meet the sides of the inscribed quadrilateral QiQ^Q^Q.i- Hence the conic has the property of the four-line locus with respect to the sides of any inscribed quadrilateral.] The beginning of Book IV. of Apollonius' work contains a series of propositions, 28 in number, in which he proves the converse of Propositions 62, 63, and 64 above for a great variety of different cases. The method of proof adopted is the reductio ad absurdum, and it has therefore been thought unnecessary to reproduce the propositions. It may, however, be observed that one of them [IV. 9] gives a method of drawing two tangents to a conic from an external point. DraAv any two straight lines through Τ each cutting the conic in two points as Q, Q' and R, R'. Divide QQ' in and RR' in 0' so that TQ:TQ' = QO: 0Q\ TR: TR' = RO' : O'R'. Join 00', and produce it both ways to meet the conic in P, P'. Then P, P' are the points of contact of the two tangents from T. INTERSECTING CONICS. Proposition 77. [IV. 24.] No two conies can intersect in snch a way that part of one of them is common to both, while the rest is not. If possible, let a portion q'Q'PQ of a conic be common to two, and let them diverge at Q. Take Q' any other point on the conmion portion and join QQ'. Bisect QQ' in 1^ and draw the diameter PV. Draw rqv(j' parallel to QQ'. Then the line through Ρ parallel to QQ' will touch both curves and we shall have in one of them qv = vq', and in the other rv = vq' ; .•. rv = qv, which is impossible. There follow a large number of propositions with regard to the number of points in which two conies can meet or touch each other, but to give all these propositions in detail would require too much space. They have accordingly been divided into five groups, three of which can be combined in a general enunciation and are accordingly given as Props. 78, 79 and 80, while indications are given of the proofs by which each particular case under all the five groups is established. The terms " conic " and " hyperbola " in the various enunciations do not (except when otherwise stated) include the double-branch hyperbola but only the single branch. The term " conic " must be understdod as including a circle. INTERSECTING CONICS. 127 Group I. Propositions depending on the more elementary considerations affecting conies. 1 . Two conies having their concavities in opposite directions will not meet in more than two points. [IV. 35.] If possible, let ABC, ADBEC be two such conies meeting in three points, and draw the chords of contact A B, BC. Then AB, BC contain an angle towards the same parts as the concavity of ABC. And for the same reason they contain an angle towards the same parts as the concavity of ADBEC. Therefore the concavity of the two curves is in the same direction : Λvhich is contrary to the hypothesis. 2. If a conic meet one branch of a hyperbola in two points, and the concavities of the conic and the branch are in the same direction, the part of the conic produced beyond the chord of contact will not meet the opposite branch of the hyperbola. [IV. 36.] The chord joining the two points of intersection will cut both the lines forming one of the angles made by the asymptotes of the double hyperbola. It will not therefore fall within the opposite angle between the asymptotes and so cannot meet the opposite branch. Therefore neither can the part of the conic more remote than the said chord. 3. If a conic meet one branch of a hyperbola, it will not meet the other branch in more points than two. [IV. 37.] The conic, being a one-branch curve, must have its concavity in the opposite direction to that of the branch which it meets in two points, for otherwise it could not meet the opposite branch in a third point [by the last proposition]. The proposition therefore follows from (1) above. The same is true if the conic touches the first branch. 4. A conic touching one branch of a hyperbola with its concave side will not meet the opposite branch. [IV. 30.] 128 THE COXICS OF APOLLONIUS. Both the conic and the branch which it touches must be on the same side of the common tangent and therefore Avill be separated by the tangent from the opposite branch. Whence the proposition follows. 5. If one branch of a hyperbola meet one branch of another hyperbola with concavity in the opposite direction in two points, the opposite branch of the first hyperbola will not meet the opposite branch of the second. [IV. 41.] The chord joining the two points of concourse will fall across one asymptotal angle in each hyperbola. It will not therefore fall across the opposite asymptotal angle and therefore will not meet either of the opposite branches. Therefore neither \vill the opposite branches themselves meet, being separated by the chord refen-ed to. 6. If one branch of a hyperbola meet both branches of another hyperbola, the opposite branch of the former will not meet cither branch of the second in two points. [IV. 42.] For, if possible, let the second branch of the former meet (»η<• branch of the latter in D, E. Then, joining DE, we use INTERSECTING CONICS. 129 the same argument as in the last proposition. For DE crosses one asymptotal angle of each hyperbola, and it will therefore not meet either of the branches opposite to the branches DE. Hence those branches are separated by DE and therefore cannot meet one another : which contradicts the hypothesis. Similarly, if the two branches DE touch, the result will be the same, an impossibility. 7. If one branch of a hyperbola meet one branch of another hyperbola with concavity in the same direction, and if it also meet the other branch of the second hyperbola in one point, then the opposite branch of the first hyperbola will not meet either branch of the second. [IV. 45.] i.1, Β being th( H. C. points of meeting ith the first branch and 9 130 THE COXJCS OF ArOLLONIUS. C that with the opposite branch, by the same principle as before, neither AC nor BC will meet the branch opposite to ACB. Also they will not meet the branch C opposite to A Β in any other point than C, for, if either met it in two points, it would not meet the branch AB, which, however, it does, by hypothesis. Hence D will be within the angle formed by AC, BC produced and will not meet C or AB. 8. If a hyperbola touch one of the branches of a second hyperbola with its concavity in the opposite direction, the opposite branch of the first will not meet the opposite branch of the second. [IV. 54.] The figure is like that in (6) above except that in this case D and Ε are two consecutive points ; and it is seen in a similar manner that the second branches of the ΐΛνο hyperbolas are separated by the common tangent to the first branches, and therefore the second branches cannot meet. Group II. containing propositions capable of being ex- pressed in one general enunciation as follows : Proposition 78. No two conies {including under the term a hyperbola with two branches) can intersect in more than four points. 1. Suppose the double-branch hyperbola to be alone excluded. [IV. 2.5.] INTERSECTING CONICS. 131 If possible, let there be five points of intersection Λ, B, C, D, E, being successive intersections, so that there are no others between. Join AB, DC and produce them. Then (a) if they meet, let them meet at T. Let 0, 0' be taken on AB, DC such that Τ A, TD are harmonically divided. If 00' be joined and produced it will meet each conic, and the lines joining the intersections to Τ will be tangents to the conies. Then TE cuts the two conies in different points P, P', since it does not pass through any common point except E. Therefore ET : Τ Ρ = ΕΙ : IP \ and ET:TP' = EI:IPy where 00', Τ Ε intersect at /. But these ratios cannot hold simultaneously ; therefore the conies do not intersect in a fifth point E. (b) If AB, DC are parallel, the conies will be either ellipses or circles. Bisect AB, DC at M, M' ; MM' is then a diameter. Draw ENPP' through Ε parallel to AB or DC, meeting MM' in Ν and the conies in P, P'. Then, since MM' is a diameter of both, NP = NE = NP', which is impossible. Thus the conies do not intersect in more than four points. 2. A conic section not having two branches will not meet a double-branch hyperbola in more than four points. [IV. 38.] This is clear from the fact that [Group I. 3] the conic meeting one branch will not meet the opposite branch in more points than two. 9—2 132 THE COXICS OF APOLLONIUS. 3. If one branch of α hyperbola cut each branch of a second hyperbola in two points, the opposite branch of the first hyperbola will not meet either branch of the second. [IV. 43.] The text of the proof in ApoUonius is corrupt, but Eutocius gives a proof similar to that in Group I. 5 above. Let HOH' be the asymptotal angle containing the one branch of the first hyperbola, and ΚΟΚ' that containing the other branch. Now AB, meeting one branch of the second hyperbola, Avill not meet the other, and therefore AB separates the latter from the asymptote OK'. Similarly DC separates the former branch from OK. Therefore the proposition follows. 4. If one branch of a hyperbola cut one branch of a second in four points, the opposite branch of the first will not meet the opposite branch of the second. [IV. 44.] The proof is like that of 1 (a) above. If Ε is the supposed fifth point and Τ is determined as before, ET meets the inter- secting branches in separate points, whence the harmonic jiroptTty produces an absurdity. 5. If one branch of a hyperbola meet one branch of a second in three points, the other branch of the first will not meet the other branch of the second in more than one point. [IV. 46.] I IXTEIISECTING CONICS. 133 Let the tirst two branches intersect in Λ, B, C\ and (if possible) the other two in D, E. Then («) if AB, DE be parallel, the line joining their middle points will be a diameter of both conies, and the parallel chord through C in both conies will be bisected by the diameter; which is impossible. (6) If AB, DE be not parallel, let them meet in 0. Bisect AB, DE in M, M', and draw the diameters MP, MP' and M'Q, M'Q' in the respective hyperbolas. Then the tangents at Ρ',Ρ will be parallel to ^0,and the tangents at Q', Q parallel to BO. L^t the tangents at P, Q and P', Q' meet in T, T'. Let CRR' be parallel io AO and meet the hyperbolas in R, R', and DO in 0'. Then TP' -.TQ'^AO.OB -.DO.OE = T'P" : T'q\ [Prop. 5i)] It follows that RO' . O'G : DO' . O'E = R'O' . O'G : DO' . O'E, whence RO' . O'G = R'O' .O'G ; which is impossible. Therefore, etc. 6. The two branches of a hyperbola do not meet the two branches of another hyperbola in more points than four. [IV. 55.] 134 THE COXICS OF APOLLONIUS. Let A, A' be the two branches of the first hyperbola and B, B' the two branches of the second. Then (a) if A meet B, B' each in two points, the proposition follows from (3) above ; (6) if A meet Β in tAvo points and B' in one point, A' cannot meet B' at all [Group I. 5], and it can only meet Β in one point, for if A' met Β in two points A could not have met B' (which it does) ; (c) if A meet Β in two points and A' meet B, A' Avill not meet B' [Group I. δ], and A' cannot meet Β in more points than two [Group I. 3] ; {d) if A meet Β in one point and B' in one point, A' will not meet either Β or B' in two points [Group I. 6] ; (e) if the branches A, Β have their concavities in the same direction, and A cut Β in four points, A' will not cut B' [case (4) above] nor Β [case (2) above] ; (/) if A meet Β in three points, A' will not meet B' in more than one point [case (5) above]. And similarly for all possible cases. Group III. being particular cases of Proposition 79. Two cunicfi {includinij duiible lijperbulas) iuhich touch at one point cannot intersect in more than two other jwints. 1. The proposition is true of all conies excluding hyperbolas with tw(j branches. [IV. 20.] The proof follows the method of Pr(^i). 78 (1) above. INTERSECTING CONICVS. 135 2. If one branch of a hyperbola touch one branch of another in one point and meet the other branch of the second hyperbola in two points, the opposite branch of the first will not meet either branch of the second. [IV. 47.] The text of Apollonius' proof is corrupt, but the proof of Prop. 78 (3) can be applied. 3. If one branch of a hyperbola touch one branch of a second in one point and cut the same branch in two other points, the opposite branch of the first does not meet either branch of the second. [IV. 48.] Proved by the harmonic property like Prop. 78 (4). 4. If one branch of a hyperbola touch one branch of a second hyperbola in one point and meet it in one other point, the opposite branch of the fii^st Avill not meet the opposite branch of the second in more than one point. [IV. 49.] The proof follows the method of Prop. 78 (5). 5. If one branch of a hyperbola touch one branch of another hyperbola (having its concavity in the same direction), the opposite branch of the first will not meet the opposite branch of the second in more than two points. [TV. 50.] The proof follows the method of Prop. 78 (.5), like the last case (4). 6. If a hyperbola with two branches touch another hyper- bola Λvith two branches in one point, the hyperbolas will not meet in more than two other points. [IV. 56.] The proofs of the separate cases follow the methods em- ployed in Group I. 3, 5, and 8. Group IV. merging in Proposition 80. No two conies touching each other at tiuo iJoints can intersect at any other point. 1. The proposition is true of all conies excluding hyperbolas with two branches. [IV. 27, 28, 29.] 136 THE COXICS OF APOLLONIUS. Suppose the conies touch at Λ, B. Then, if possible, let them also cut at G. (a) If the tangents arc not parallel and C does not lie between A and B, the proposition is proved from the harmonic property ; (6) if the tangents are parallel, the absurdity is proved by the bisection of the chord of each conic through G by the chord of contact which is a diameter ; (c) if the tangents are not parallel, and G is between Λ and B, draw TVirom the point of intersection of the tangents to the middle point of ΛΒ. Then TV cannot pass through G, for then the parallel through G to ΛΒ would touch both conies, which is absurd. And the bisection of the chords parallel to A Β through G in each conic results in an absurdity. 2. If a single-branch conic touch each branch of a hyper- bola, it will not intersect either branch in any other point, [IV. 40.] This follows by the method employed in Group I, 4. 3. If one branch of a hyperbola touch each branch of a second hyperbola, the opposite branch of the first will not meet either branch of the second. [IV. 51.] Let the branch AB touch the branches AG, BE in A, B. Draw the tangents at ^, J5 meeting in T, If possible, let GD, the opposite branch to AB^ meet AG in G. Join GT. Then Τ is within the asymptotes to AB, and therefore GT falls within the angle ATB. But BT, touching BE, cannot meet the opposite branch AC. Therefore BT falls on the side of GT remote from the branch AG, or GT passes through the angle adjacent to A TB ; which is impossible, since it foils withiTi the angh- ATB. INTEUSECTING CONICS. 137 4. If one branch of i)ue hyperbola touch one branch of another in one point, and if also the other branches touch in one point, the concavities of each pair being in the same direction, there arc no other points of intersection. [IV. 52.] This is proved at once by means of the bisection of chords parallel to the chord of contact. 5. If one branch of a hyperbola touch one branch of another in two points, the opposite branches do not intersect. [IV. 53.] This is proved by the harmonic property. 6. If a hyperbola with two branches touch another hyper- bola with two branches in two points, the hyperbolas will not meet in any other point. [IV. 57.] The proofs of the separate cases follow those of (3), (4), (5) above and Group I. 8. Group V. Propositions respecting double contact bet\vcon conies, 1. Λ parabola cannot touch another parabola in more points than one. [IV. 30.] This follows at once from the property that TP = Ρ V. 2. A parabola, if it fall outside a h}^erbola, cannot have double contact with the hyperbola. [IV. 31.] For the hyperbola CV:CP = CP:CT = GV-CP:CP-CT = PV:PT. Therefore PV>PT. And for the parabola P'V=P'T: therefore the hyperbola falls outside the parabola, which is impossible. 3. A parabola cannot have internal double contact with an ellipse or circle. [IV. 32] The proof is similar to the preceding. 1:38 THE COXICS OF APOLLONIUS. 4. A hyperbola cannot have double contact with another hyjxirbola having the same centre. [IV. 33.] Proved by means oiGV.CT= CP\ 5. If an ellip.se have double contact Avith an ellipse or a circle having the same centre, the chord of contact will pass through the centre. [IV. 34.] Let (if possible) the tangents at A, Β meet in T, and let V be the middle point of AB. Then TV is a diameter. If possible, let G be the centre. Then CP^= CV. GT=CF\ which is absurd. Therefore the tangents at ^, 5 do not meet, i.e. they are parallel. Therefore AB '\& Ά diameter and accordingly passes through the centre. NORMALS AS MAXIMA AND MINIMA. Proposition 81. (Preliminary.) [V. 1, 2, 3.] If in an ellipse or a hyperbola AM he d7'awn perpendicular to the aa;is A A' and equal to one-half its parameter, and if CM meet the ordinate PN of any point Ρ on the curve in H, then PN' = 2 (quadrilateral ΜΑΝΗ). Let AL be twice AM, i.e. let AL be the latus rectum or parameter. Join A'L meeting PN in R. Then A'L is parallel to CM. Therefore HR = LM = AM. Now PN"" = AN. NR ; [Props. 2, 3] .•. PN' = AN(AM + HN) = 2 (quadrilateral ΜΑΝΗ). In the particular ca.sc where Ρ is between C and A' in the fuKJvz .,. , 140 THE coyjcs υι•' apolloxius. ellipse, the ([uadrilateral becomes the difference between two triangles, and P'N" = 2 ( Δ CA Μ - Δ CN'H ' ). Also, if Ρ be the end of the minor axis of the ellipse, the quadrilateral becomes the triangle CAM, and BC'^2ACAM. [The two l;ist cases are proved by Apollonius in separate pruptisitions. Cf. the note on Prop. 23 above, p. 40.] Proposition 82. [V. 4.] 7/i a pardbola, if Ε he a point on the axis such that AE is e(jual to half the latus rectum, then the minimum strairjht line from Ε to the curve is AE ; and, if Ρ he any other point on the curve, PE increases as Ρ moves further from A on either side. Also for any point PE'=AE' + AN-\ Let AL ho the parameter or latus rectum. Then PN* = AL.AN = 2AE.AN. Adding EN*, we have PE'=2AE.AN+EN' =^2AE.AN + (AE'- ANf =^AE'+AN\ NORMALS AS MAXIMA AND MINIMA. 141 Thus PE'^ > AE' and increases with AN, i.e. as Ρ moves further and further from A. Also the minimum value of PE is AE, or AE is the shortest straight line from Ε to the curve. [In this proposition, as in the succeeding propositions, Apollonius takes three cases, (1) where Ν is between A and E, (2) where Ν coincides with Ε and PE is therefore perpen- dicular to the axis, (3) where AN is greater than AE, and he proves the result separately for each. The three cases will for the sake of brevity be compressed, where possible, into one.] Proposition 83. [V. 5, G.] If Ε he a point on the axis of a hyperbola or an ellipse such that AE is equal to half the latus rectum, then AE is the least of all the straight lines which can he draimi from Ε to the curve; and, if Ρ he any other point on it, PE increases as Ρ moves further from A on either side, and PE" = AE' + AN' . ^4^^ [= ^E" + e' • ^N'] AA '■ {luhere the upper sign refers to the hyperhola)*. Also in the ellipse Ε A' is the maximum straight line from Ε to the curve. Let AL be draΛvn perpendicular to the axis and equal to the parameter; and let^X be bisected at if, so that^iT/= J.^". Let Ρ be any point on the curve, and let PN (produced if necessary) meet CM in Η and EM in K. Join EP, and draw MI perpendicular to HK. Then, by similar triangles, MI = IK, and EN = NK. * The area represented by the second term on the right-hand side of the equation is of course described, in Apollonius' phrase, as the rectangle on the base .-Ιλ'' similar to that contained by the axis (as base) and the sum (or difference) of the axis and its parameter. A similar remark applies to the similar expression on the next page. 142 THE CONK'S OF APOLLONIUS. Now PN^ = 2 ((luadrilateral ΜΑΝΗ), and λ\\'=2ΑΕΧΚ; .•. PE'=2(AEAM+ AMHK) = AE' + MI.HK = AE' + MI.(IK±IH) = ΑΕ' + ΜΙ.{3Π±ΙΗ).... [Prop. 81] (1)• κ (<i ^ ^ >^ ^ ^c y — l^ Now .1// : IE = CA : AM = yl^' : ,ρ^. Therefore MI . (ili/ ± IH) : ^^' . (^^' ± Pa) = i»^/' : A A' MP MI.{MI±IH) = ^^,.AA'.{AA' ±pa) - MT^ AA'±Pa ~^'^ ' AA' AA' ± Pa • AA' Avhence, by means of (1), PE' = AE' + AN' . ^^' 7,^" . AA' It follows that AE is the minimum value of PE, and that PE increases with AN, i.e. as the point Ρ moves further from A. Also ill the ellipse the mcucimum value οι PE' is AE' + AA' {AA' - Pa) = AE' + A A" -2AE. A A' = EA"\ NORMALS AS MAXIMA Α\Ό MINIMA. 143 Proposition 84. [V. 7.] If any point be taken on the a:cis of any conic such that AO < hpa, then OA is the minimum straight line from to the cin-ve, and OP (if Ρ is any other point on it) increases as Ρ moves further and furtlier from A. Let AEhQ set off along the axis equal to half the parameter, ami join PE, PO, PA. Then [Props. 82, 83] PE > AE, so that δΡΑΕ>δΑΡΕ\ and a fortiori δΡΑΟ>δΑΡΟ, so that PO>AO. And, if P' be another point more remote from A, P'E > PE. .•. ZEPP'>ZEP'P; and a fortioH Ζ OPP' > Ζ OF P. .•. OP'>OP, and so on. Proposition 85. [V. 8.] I7i a imrahola, if G he a point on the axis such that AG>\pa, inid if Ν be taken between A and G such that NG 2' then, if NP is dravm perpendicidar to the axis meeting the curve in P, PG is the minimum straight line from G to the carve [or the normal at P]. 144 THE COXICS OF APOLLONTUS. // F' be any other jymnt on the curve, P'G increases as P' mon furthei' from Ρ in either direction. Also P'G' = PG'-\-NN'\ Wchave P'N"=pa-AN' = 2NG.AN'. Also N'G' = NN'^ + NG' ± 2NG . NN' (caccording to the position of N'). Therefore, adding, P'G' = 2NG .AN+ NN'' + NG' = PN' + NG' + NN" = PG' + NN''. Thus it is clear that PG is the minimum straight line from G to the curve [or the normal at P]. And P'G increases with NN', i.e. as P' moves further from /■* in either direction. Proposition 86. [V. 9, 10, 11.] /// a hyperbola or an ellipse, if G be any point on Λ A' (within the curve) such that AG>^, and if GN be measured towards the nearer veiiex A so that NG :CN = pa:A A' [= CB' : CA'], yORMALS AS MAXIMA AND MINIMA. 145 then, if the ordinate through Ν meet the curve in P, PG is the minimum straight line from G to the curve [or PG is the nonnal at P] ; ai^d, if P' be any other point on the curve, P'G increases as P' moves further from Ρ on either side. Also P'G' - PG' = NN" . "^-4^4^ ΑΛ [=e\NN' where P'N' is the ordinate from P'. f / <| Κ jf c A / \ Ψ' Ν N- / /g ^ \^ / <;. Ρ^"Ν k 1 >-- ,H" /I "^ 'y^^ ?^ \ ■■' ^^\ / Ν \ .;k' \ / \ V 1 V Ν' Ν A- yc ;n" ^ Η ?\ ^^ ^^ Draw AM perpendicular to the axis and equal to half the parameter. Join CM meeting PN in Η and P'W in K. Join GH meeting P'N' in K'. Then since, by hypothesis, NG'.GN = pa'.AA', and, by similar triangles, NH'.GN = AM:AQ = Pa '-ΛΑ', it follows that NH = NG, whence also N'H' = N'G. Now PN'=2 (quadrilateral ΜΑΝΗ), [Prop. 81] NG' = 2AHiYG. Therefore, by addition, PG"" = 2 (quadrilateral AMHG). H. c. 10 146 THE coyics of apolloxius. Also P'G' = FN" + N'G* = 2 (quadr. AMKN') + 2 Δ H'N'G = 2 (quadr. AMHG) + ^CsEH'K. PG''='2/\HH'K = HI .{H'I±IK) = HI. {HI ± IK) P'G' = HP CA ± AM ' GA NN' Thus it follows that PG is the minimum straight line from G to the curve, and P'G increases with NN' as P' moves further from Ρ in either direction. In the ellipse GA' will be the maanmum straight line from G to the curve, as is easily proved in a similar manner. Cor. In the particular case where G coincides Avith C, the centre, the two minimum straight lines are proved in a similar manner to be CB, CB', and the two maxima CA, CA', and CP increases continually as Ρ moves from Β to A. Proposition 87. [V. 12.] If G be a point on the axis of a conic and GP be the mini- mum straight line from G to the curve \or the normal at P\ and if be any point on PG, then OP is the minimum straight line from to the cui^je, and OP' continually increases as P' moves from Ρ to A [or to A']. Since FG > PG, zGPP'>zGPP. NORMALS AS MAXIMA ANT) MINIMA. 147 Therefore, a fortimn, Ζ OP Ρ' > Ζ OF Ρ, or OP' > OP. Similarly OP" > OP' [&c. as in Prop. 84]. [There follow three propositions establishing for the three curves, by red actio ad crbsurdum, the convei-se of the propo- sitions 85 and 86 just given. It is also proved that the normal makes with the axis towards the nearer vertex an acute angle.] Proposition 88. [Y. 16, 17, 18.] If E' be a point on the minor axis of an ellipse at a distance GA' then E'B from Β equal to half the parameter of BE' or ^„ is the maximum straight line from Ε to the curve ; and, if Ρ he any other point on it, E'P diminishes as Ρ moves further from Β on either side. Also E'B'-E'P^Bn'.'' -η^ [=£«'. '^^] . ApoUonius proves this sepai-ately for the cases (1) where ^<BB', (2) whei-e ^=BB', and (3) where ^>BB'. The method of proof is the same for all three cases, and only the first case of the three is given here. *'"^"""'~?<"' "vT" ;,■'>■ -----y<• -■ 10—2 148 THE CONICS OF APOLLONIUS. By Prop. 81 (which is applicable to either axis) we have, if Bm =^ = BE', and Pn meets Cm, E'm in h, k respectively, P/i'= 2((iua(lnlatcral mBnh). Also ηΕ"=2Α»1•Ε'. .'. PE'^=2AmBE'-2Amhk. But BE'*=1AviBE'. .•. BE"-PE"=^2Amhk = mi . (hi — ki) = mi . (hi — mi) ^ mB-CB whence the proposition folloAVS. Proposition 89. [V. 19.] If BE' be measured along the minor axis of an ellipse equal Γ CA'^~\ to half the jiciraineter or γ^ and any point be taken on the minor axis such that BO > BE', then OB is the maximum straight line from to the curve; and, if Ρ be any otJier point on it, OP diminishes continually as Ρ moves in either direction from Β to B'. The proof follows the method of Props. 84, 87. NORMALS AS MAXIMA AX I) MINIMA. 149 Proposition 90. [V. 20, 21, 22.] If g he a point on the minor axis of an ellipse such that or γψ^ \ , and if Gn he measured to- luards Β so that Cn:ng = BB':p^[=CB':CA'l then the perpendicular through η to BB' will meet the curve in two points Ρ such that Pg is the maximum straight line from g to the curve. Also, if P' he any other point on the curve, P'g diminishes as P' moves further from Ρ on either side to Β or B', and Pg' rg -nn . ^^, ,, CA'-CBn A^<' n' h'yf\ \ Draw Bm perpendicular to BB' and equal to half its para- meter pi,. Join Cm meeting Pn in h and P'n in h', and join gh meeting P'n in k. Then since, by hypothesis, Cn :ng = BB'.pb = BC: Bm, and Cn : nh = BC : Bm, by similar triangles, it follows that ng = nh. Also gn = n'k, and hi = ik, where hi is perpendicular to P'n. 150 THE OONICS OF APOLLONIUS. Now Pn^= 2 (quadrilateral mBnh), ng^ = 2A}ing: .•. Pg'=2(mBnJi + Ahng). Similarly P'g' = 2 {mBn'h' + Δ hi'g). By subtraction, Pg^-Py=2/S},h'l• = hi . (h'i — ki) = hi.{h'i — hi) [Em - BC\ hi' V BG -nn . ^^ , whence it follows that Pg is the maximum straight line from g to the curve, and the difference between Pg^ and P'g^ is the area described. Cor. 1. It follows from the same method of proof as that used in Props. 84, 87, 89 that, if be any point on Pg produced beyond the minor axis, PO is the mammum, straight line that can be drawn from to the same part of the ellipse in which Pg is a maximum, i.e. to the semi-ellipse BPB', and if OF be drawn to any other point on the semi-ellipse, OP' diminishes as P' moves from Ρ to Β or B'. Cor. 2. In the particular case where g coincides with the centre C, the maximum straight line from C to the ellipse is perpendicular to BB', viz. CA or GA'. Also, if g be not the centre, the angle PgB must be acute if Pg is a maximum ; and, if Pg is a maximum [(jr a normal], (hi: ng = GB': GA\ [This corollary is proved separately by redmtio ad absurdum.] NORMALS AS MAXIMA AND MINIMA. 151 Proposition 91. [V. 2.S.] If g he on tlie minor axis of an ellipse, and gP is a nicucimum straight line from g to the curve, and if gP meet the major axis in G, GP is a minimum straight line from G to the cin've. [In other words, the minimum from G and the maximum from g determine one and the same normal.] Φ We have Cn : ng = BB' : pb [Prop. 90] [= CB'' : CA'] = p„: A A'. Also Gn : ng = PN : ng = NG : Pn, by similar triangles. = NG : CN. .•. NG'.CN=pa:AA', or PG is the normal determined as the minimum straight line from G. [Prop. 86] Proposition 92. [V. 24, 25, 26.] Only one normal can be drawn from any one point of a conic, whether such normal be regarded as the minimum straight line from the point in which it meets A A', or as the maximum straight line from the point in which (in the case of an ellipse) it meets the minor axis. 152 THE (JOXICS OF APOLLONIUS. This is at once proved by reductio ad ahsiirdum on assuming that PG, Ρ Η (meeting the axis A A' in G, H) are minimum .straight lines from G and Η to the curve, and on a similar assumption for the minor axis of an ellipse. Proposition 93. [V. 27, 28, 29, 30.] The nonmil at any point Ρ υη a conic, whetJter regarded as a minimum straight line from its intei'section with the axis ■ A A' or as a maximum from its intersection with BE (in the ' case of an ellipse), is perpendicular to the tangent at P. Let the tangent at Ρ meet the axis of the parabola, or the axis A A' οι Ά hyperbola or an ellipse, in T. Then we have to prove that TPG is a right angle. (1) For the parabola wo have ΑΤ = Αλ^, and NG = ^', .•. NG : pa = AN : NT, •so that TN.NG=pa.AN = PN\ And the angle at Ν is a right angk• ; .•. Ζ TPG is a right angle. I NORMALS AS MAXIMA AND MINIMA. 158 (2) For the hyperbola or ellipse PN':CN.NT = Ρα•.ΑΑ' [Prop. U] = NG : CN, by the property of the minimum, [Prop. 86] = TN.NG:CN.NT. .•. PN^ = TN.NG, while the angle at Ν is right ; .•. Ζ TPG is a right angle. (3) If Pg be the maximum straight line from g on the minor axis of an ellipse, and if Pg meet Λ A' in G, PG is a minimum from G, and the result follows as in (2). [Apollonius gives an alternative proof applicable to all three conies. If GP is not perpendicular to the tangent, let GK be perpendicular to it. Then Ζ GKP > ζ GPK, and therefore GP > GK. Hence a fortiori GP > GQ, where Q is the point in which GK cuts the conic; and this is impossible because GP is a minimum. Therefore &c.] Proposition 94. [V. 31, 33, 34.] (1) In general, if be any point luithin a conic and OP be a maadmum or a minimum straight line from to the conic, a straight line PT drawn at right angles to PO will touch the conic at P. 154 THE COXICS OF APOLLONIUS. (2) If 0' be any point on OP produced outside the conic, then, of all straight lines drawn from 0' to meet the conic in one point but not produced so as to meet it in a second point, O'P vnll be tlie minimum; and of the rest that which is nearer to it will be less than that which is more remote. (1) First, let OP be a maocimum. Then, if Τ Ρ does not touch the conic, let it cut it again at Q, and draw OK to meet PQ in Κ and the curve in R. i Then, since the angle OPK is right, Ζ OPK > Ζ OKP. Therefore OK > OP, and a fortiori OR > OP : which is impossible, since OP is a maximum. Therefore TP must touch the conic at P. Secondly, let OP be a minimum. If possible, let TP cut the curve again in Q. From any point between Τ Ρ and the curve draw a straight line to Ρ and draw ORK perpendicular to this line meeting it at Κ and the curve in R. Then the angle OKP \h a right angle. Therefore OP > OK, and a fortiori OP > OR : which is impossible, since OP is a minimum. Therefore TP must touch the curve. NORMALS AS MAXIMA Α\Π MINIMA. 155 (2) Let 0' be any point on OP produced. Dmw the tangent at P, as PK, which is therefore at right angles to OP. Then draw O'Q, O'R to meet the curve in one point only, and let O'Q meet PK in K. Then O'K > O'P. Therefore a fortiori O'Q > O'P, and O'P is a minimum. Join RP, RQ. Then the angle O'QR is obtuse, and therefore the angle O'RQ is acute. Therefore O'R > O'Q, and so on. Proposition 95. [V. 35, 86, 37, 38, 39, 40.] (1) If the normal at Ρ meet the lucis of a parabola or the axis ΛΛ' of a hyperbola or ellipse in G,the angle PGA increases as Ρ or G moves further and further from A, but in the hyperboL• the angle PGA will ahuays be less than the complement of half the angle betiueen the asymptotes. (2) Tiuo normals at points on the same side of the a-xis AA' will meet on the opposite side of that axis. (3) Two normals at points on tJie same quadrant of an ellipse, as AB, will meet at a point luithin the angle ACB'. (1) Suppose P' is further from the vertex than P. Then, since PG, P'G' are minimum straight lines from G, G' to the curve, we have .56 THE (JOXJCS OF APOLLONIUS. (a) For the parabola and Γ'Ν'>ΡΝ\ δΡΤτ'Α> δΡΘΑ. ρ 3 7 / ^\ Ν κ Ν, \ > "-"έ?:. C ) 9 / 0' / (6) For the hyperbola and ellipse, ]οι\\\\\^ CP and producing it if necessary to meet P'N' in K, and joining KG', we have NV : CN'=pa : AA' [Prop. 86] = NG:GN; .-.N'G' :NG=GN' -.GN = KN' : PN, by similar triangles. Therefore the triangles PNG, KN'G' are similar, and ^KG'N'=lPGN. Therefore Ζ P'G'N' > Ζ PGN. (c) In the Jirjperbola, let AL be drawn perpendicular to A A' to meet the asymptote in L and GP in 0. Also let AM be ecpial to ^ . Now AA' •.2ya = GA:AM=GN'.NG, and CM : GA = PN : GN, by similar triangles ; therefore, ex aequali, OA : AM = PN : NG. Hence AL : AM > PN : NG. NORMALS AS MAXIMA AND MINIMA. 1ό7 But AL : AM= CA : AL , [Prup. 2.S] .•. CA .AL>PN:NG\ :. Δ PGN is less than Ζ CLA. (2) It follows at once from (1) that two normals at points on one side of A A' \ή\\ meet on the other side of A A'. (3) Regard the two normals as the maximum straight lines from g, (/', the points where they meet the minor axis of the ellipse. Then On : n'g' = BE : pi, [Prop. 90] = Cn : ng ; .•. On' : Cg = On : Gg. But On >0n•, .•. Og > Og, whence it follows that Pg, P'g' must cross at a point before cutting the minor axis. Therefore lies on the side of BB' toAvards A . And, by (2) above, lies below AG; therefore lies within the ΔΑΟΒ'. Proposition 96. [V. 41, 42, 43.] (1) In a parabola or an ellipse any normal PG will meet the cu?-ve again. (2) In the hyperbola (a), if AA' he not greater than pa, no normal can meet the curve in a second point on the same branch ; but (b), if AA'>pa, some normals luill meet the same branch again and others not. (1) For the ellipse the proposition is sufficiently obvious, and in the parabola, since PG meets a diameter (the axis), it will meet another diameter, viz. that through the point of contact of the tangent parallel to PG, i.e. the diameter bisecting it. Therefore it will meet the curve again. 1ό8 THE voyics of apollonius. (2) (a) Let CL, CL be the asymptotes, and let the tangent at A meet them in L, L . Take AM equal to ~. Let FO be any normal and FN the ordinate. Then, by hypothesis, CA -^ AM, and CA : AM = CA' : ΑΓ ; [Prop. 28] .•. CA If^AL; hence the angle CLA is not greater than ACL or ACL'. But Ζ CZ^ > ζ PGiV ; [Prop. 95] .. /.ACL'>ZFGN. It follows that the angle ACL' together with the angle adjacent to FON will be greater than two right angles. Therefore FO will not meet CL towai'ds L' and therefore will not meet the branch of the hyperbola again. (b) Suppose C^ > ^il/ or ^ . Then LA ■.AM>LA .AC. Take a point Κ on AL such that KA -.AM^LA : AC, NORMALS AS MAXIMA AND MINIMA. 159 Join CK, and produce it to meet the hvperbola in P, and let PN be the ordinate, and PG the normal, at P. PG is then the minimum from G to the curve, and NG ■.CN=pa:AA' =AM:Aa Also CN : PN=AC : AK.hy similar triangles. Therefore, ex aequali, NG : PN = AM : A Κ = CA : AL, from above. Hence ^ACL'=Z ACL = Ζ PGN; .•. PG, CL' are parallel and do not meet. But the normals at points between A and Ρ make with the axis angles less than the angle PGN, and normals at points beyond Ρ make with the axis angles greater than PGN. Therefore normals at points between A and Ρ will not meet the asymptote CL', or the branch of the hyperbola, again ; but normals bevond Ρ λυϊΙΙ meet the branch again. 160 THE CONICS OF APOLLONIUS. Proposition 97. [V. 44, 45, 46, 47, 48.] If Pfi^, Ρβ^ he nornuds at points on one side of the cucis of a conic meeting in 0, and if he joined to any othei' point Ρ on the conic (it heincf further supposed in the case of the ellipse that (ill three lines OP^, OP^, OP cut the same half of the aads), then y 1 ) OP cannot he a normal to the curve ; (2) if OP meet the axis in K, and PG he the normal at P, AO < AK when Ρ is intermediate between P, and P^, • and AG> AK when Ρ does not lie hetiueen P^ and P^. I. First let the conic be a parabola. Λ^ν,• Let P^P^ meet the axis in T, and draw the ordinatcs P,-A^,, NORMALS AS MAXIMA AM) MINIMA. 161 Draw OM perpendicular to the axis, and measure MH towards the vertex equal to ^ . Then MH = A\G„ and N^H=G,M. Therefore MH : HN.^ = Χβ, 4- G^M = PjiVjj : MO, by similar triangles. Therefore HM .ΜΟ = Ρ,Ν^.Νβ) Similarly HM .M0 = Ρ^Ν^.Νβ] ^ ^' Therefore Ηλ\ : Ηλ\ = P^N^ : P^N^ whence N^h\ : HN^ = Λ\Ν^ : TN^ ; and TN^ = HNJ ^ ^" If Ρ be a variable point and PN the ordinate*, Ave have now three cases : TN<TN^ or HiY^ (1), TN>TN^ or ^^Y,, but < TN^ or HN^ (2), TN>TN^ or HN^ (3). Thus, denoting the several cases by the numbers (1), (2), (3i, we have N,N'.TN>N^N:HN^ (1), <N,N:HN, (2), <N,N:HN, (3), and we derive respectively TN^:TN>HN:HN^_ (1), <HN'.HN.^ (2), >HN:HN^ (3). * It will be obser\-ed that there are three sets of points P, N, K, in the figure denoted by the same letters. This is done in order to exhibit the three different cases ; and it is only necessary to bear in mind that attention must be confined to one at a time as indicated in the course of the proof. H. c. 11 162 THE COXICS OF APOLLONIUS. If NP meet P^P^ in F, we have, by similar triangles, P^N, : FX>HN : HN, (1) and (8), <HN.HN, (2). But in (1) and (3) FN > PN, and in (2) FN < PN Therefore, a fortiori in all the cases, Ρβ^ : PN>HN : HN, (1) and (3), <HN:HN, (2). Thus P^N^.N,H>PN.NH (l)and (3), <PN.NH (2). Hence HM. MO > PN.NH... (!) and (S)) .,..•, <PN.NH ^^^\,hy(A)aho.e. Therefore MO ■.PN>NH:HM (1) and (3), <NH:HM (2) and MO'.PN = MK:NK. Therefore MK -. NK> NH : HM (1 ) and (3), <NH:HM (2), whence we obtain MN : NK > MN : Η Μ (1) and (3), <MN:HM (2), 80 that HM or NG > NK in (1) and (3), and < NK in (2). Thus the proposition is proved. II. Let the conic be a hyperbola or an ellipse. Let the normals at Pj , P.^ meet at 0, and draw OM perpen- dicular to the axis. Divide CM in Η (internally for the hyperbola and externally for the ellipse) so that CH : HM = AA' : pa [or CA' : CB'], and let OM be similarly divided at L. Draw HVR parallel to OM and LVE, Oi?P parallel to CM. NORMALS AS MAXIMA AND MINIMA. 1G.3 Suppose P..Pi produced to meet EL in T, and let FiN^, P.N. meet it in U„ U.. Take any other point Ρ on the curve. Join OP meeting the axes in K, k, and let Ρ Ν meet P^P. in Q and EL in U. 11—2 1G4• THE coxirs of apollonius. Now OiY. : N,G, = ΛΛ' ■.p„ = GH: HM. Therefore, componendo for the hyperbohx and dividendo for the ellipse, CM:GH=GG.:GN^ = GG,~GM:GN,-GH = MG,'.HN, = MG,: VU, (A). Next FE : EG=AA' : pa = GN^ : Ν.β,, so that FG:GE = GG, : NM,. Thus FG:N,U,==GG, : N,G, = Gg. : P.>N.., by similar triangles, = FG±Gg,'.N.JJ,±PJ(, = FgS':PJJ, Γ! (B). Again FG. GM : EG. GH = {FG : C^) . {GM : CiO = {Fg,:PM.;).{MG,: VU,), from (A) and (B), and FG . GM = Fg, . MG, , '.• Fg,: GM = FG : il/(?o . .•. EG.GH = P,U,.U,V, or GE.EV=PM,.U,V = PJJi. Ui V, in like manner ; .•. L\V: U,V=PM,:P,U, = TU., : TUi, by similar triangles, whence U,U, : U,V= U,U,: TU, ; .:TU,= VU,l .^. and TU,= VUj ^ ^' Now suppose (1) that AN < AN'^; then t/^,F > TU, from (C) above ; .•. UU,:TU>UU,:U,V; hence Τίλ^: ΓΙ7 > i7F : /7,F; ••• 2\U^.QU>UV: UJ, by similar triangles. Therefore PJ\^. UJ>QU. UV, Μ\Λ a fortiori >PU.UV, NORMALS AS MAXIMA AND MINIMA. 165 But 1\^ ϋ\ .1\ν= CE . Ε V, iVuin above•, = LO.OR, •.• CE.LO = uR.EV; .•. LO.OR>FU.UV. Suppose (2) that yliY>^iY, but < xiN^. Then TU^ < UV; .•. U^U:Tl\>l\U: UV, whence TU:TU^> U^V : UV ; Λ QU:P^U^>U^V: UV, by similar triangles. Therefore {a fortiori) PU . UV >P^U^.UJ >LO.OR Lastly (3) let AN be > AN,. Then TU^ > UV; .•. U^U:TU^< U,U: UV, whence TU:TU^< UJ: UV, or QU:P^U^< U^V : UV; and afortioH >PU . UV\ .•. LO.uR>PU.UV, as in (1) above. Thus we have for cases (1) and (3) LO.OR>PU.UV, and for (2) LO.OR<PU. UV That is, we shall have, supposing the upper symbol to refer to (1) and (3) and the lower to (2), LO-.PU^ UV:OR, i.e. LS-.SU^ UV:LV; .•. LU: US^LU-.LV, and LV^US. 166 THE COXICS OF APOLLONIUS. It follows that FO'.LV^FO: SU, or Fk : PU, or CM:MH^Fk:PU', .•. FC: CE^Fk:PU ^FkTFC:PU+CE J Ck : Ρ Ν J CK : i\r/f . Therefore, componendo or dividendo, FE : jB:6' ^ CiV^ : iVZ, or CN : NK^FE: EC, i.e. 2^^':j)„. But (7i\r:i\r(; = yl^':^^; .•. NK^NG; i.e, when Ρ is not between P^ and P^ NK< NG, and when Ρ lies between P, and P.^, NK>NG, whence the proposition follows. Cor. 1. In the particular case of a quadrant of an ellipse where P, coincides \vith B, i.e. Avhere coincides with g^, it follows that no other normal besides P,f/i, Bg^ can be drawn through g^ to the quadrant, and, if Ρ be a point between A and P, , while Pg^ meets the axis in K, NG > NK. But if Ρ lie between P, and P, iVG < NK. [This is separately proved by ApoUonius from the property in Prop. 95 (8).] C(JU. 2. 77<?-ee normals at i^oints on one quadrant of an ellipse cannot meet at one point. This follows at once from the preceding propositions. { NORMALS AS MAXIMA AND MINIMA. 167 Cor. 3. Four 7ioi'mals at points on one semi-ellipse bounded by the major axis cannot meet at one point. For, if four such normals cut the major axis and meet in one point, the centre must (1) separate one normal from the three others, or (2) must separate two from the other two, or (3) must lie on one of them. In cases (1) and (3) a contradiction of the preceding proposition is involved, and in case (2) a contradiction of Prop. 90 (3) which requires two points of intersection, one on each side of the minor axis. Proposition 98. [V. 49, 50.] In amj conic, if Μ be any point on the axis such that AM is not greater than half the latus rectum, and if be any point on the perpendicular to the axis through M, then no straight line drawn to any point on the curve on the side of the axis opposite to and meeting the axis between A and Μ can be a normal. Let OP be draAvn to the curve meeting the axis in K, and let PN be the ordinate at P. We have in the parabola, since AM'i^^ NM<^, i.e.<NG. Therefore, a fortiori, Ν Κ < NG. For the hyperbola and ellipse AA' : jh is not greater than CA : AM, and CN:NM>CA :AM; .•. CN : NM > AA' : pa >CN:NG; .•. NM<NG, and a fortiori λΊ\: < NG. Therefore OP is not a normal. PROPOSITIOXS LEADING IMMEDIATELY TO THE DETERMINATION OF THE Ε VOLUTE. Proposition 99. [V. 51, 52.] If AM measured along the axis he greater than ^ {but in the case of the ellipse less than AC), and if MO be drawn 2)erj)endicular to the a^is, then a certain length [?/] can be assigned such that (a) if OM > y, no normal can be drawn through which cuts the axis ; hut, if OP be any straight line draiun to the curve cutting the a.ds in K, NK< NG, where Ρ Ν is the ordinate and PG the normal at Ρ ; (b) if OM=y, only one normal can he so drawn through 0, and, if OP he any other straight line drawn to the curve and meeting the axis in K, Ν Κ < NG, as before ; (c) if OM < y, two normals can be so draiun through 0, and, if OP he any other straight line drawn to the curve, NK is less or greater than NG according as OP is not, or is, inter- mediate between the two iiornials. I. Suppose the conic is a parabola. Measure MH towards t at .V, so tlmt //.V, = 2.Y,.4 Measure MH towards the vertex equal to §, and divide AH PR()P(JSITI()NS DETEHMIXIXG THE EVoLVTE. Take a length y such that where P^N^ is the ordinate passing through iV,. (a) Suppose OM > y. 160 Join QP^ meeting the axis in K^ . Then y.P^N^ = N^H.HM\ .•. OM:P^N^>N^H:HM, or MK^ : K^N^ >N^H: HM ; henee il/iV, : iY,/i^, > il/i\r^ : HM, so that iVjii, < HM, i.e. iV^A<f• Therefore OP^ is not a normal, and N^K^ <N^G^. Next let Ρ be any other point. Join OP meeting the axis in K, and let the ordinate PN meet the tangent at P, in Q. 70 Then, if ^iV< AN^ , avc have, since Λ\Τ=2Λλ\ = Λ\Η, λ\Η>ΝΤ; thus TN^:TN>HN.HN^, or P^N^: QN>HN:HN^, and a fortiori or P^N^.NJI>PN.NH', But THE COyiCS OF APOLLONIUS If AN>AN^, NJ>NH; .'. N^N:NH>N^N:NJ, whence HN^ \HN>TN: TN^ > QN : P,N^ >PN:P^N^, a fortiori .'. P^N^.N^H>PN.NH. OM . Μ Η > P^N^ . N^H, by hypothesis ; OM.MH>PN.NH, or OM.PN>NH.HM, i.c. MK:KN>NH:HM, by similar triangles. Therefore, componendo, MN : NK > MN : HM, whence NK < HM ov ^ . Therefore OP is not a normal, and Ν Κ < NG. (b) Suppose OM = y, and Λνβ have in this case MN. '.NK=MN, .HM, N.G. or N^K^ = HM= ^ and P,0 is a normal. If Ρ is any other point, we have, as before, P,N^.N^H>PN.NH, and PjiYj . N^JI is in this case equal to OM . MH. Therefore OM . MH > PN . NH, and it follows as before that OP is not normal, and NK < NG. (c) Lastly, if J/ < 7/, OM:P^N^<N^H:HM, or OM.MH<P^N^.N,H. Let N^li be measured along iV,P, so that OM.MH=RN,.N,H PROPOSITIONS DETERMINING THE EVOLVTE. 171 Thus R lies within the curve. Let HL be drawn perpendicular to the axis, and with AH, HL as asymptotes draw a hyperbola passing through R. This hyperbola will therefore cut the parabola in two points, say P, P'. Now, by the property of the hyperbola, PN.NH = RN^.N^H = OM . MH, from above ; .•. OM:PN = NH '.HM, or MK : KN = NH : HM, and, componendo, MN : NK = MN : HM ; .•. NK=HM=^^ = NCr, and PO is normal. Similarly P'O is normal. Thus we have two normals meeting in 0, and the rest of the proposition follows from Prop. 97. [It is clear that in the second case where OM=y, is the intersection of two consecutive normals, i.e. is the centre of curvature at the point P^. If then x, y be the coordinates of 0, so that AM=x, and if 4a=^j„, HM=2a, N^H = l{x-1a\ AN^ = ^{x- 2a). Also y':P^N^'=N^H':HM\ or y':^a.AN^ = N^H' :4a'; .•. af = AN,.N^H' = ^\{x-2a)\ or 27(/ i/' = 4 (.r - 2(0', which is the Cartesian e([uation of the evolute of a parabola.] 172 THE CUXIC.S OF APOLLONIUS. II. Lol the curve be a HYPERBOLA or lui ELLIPSE. Wo have AM > -^^ , so that CA : AxM< AA' : jh- Q fi Q / 1?' N' Η \K' Μ / A Ν Ν, tt "I l(> Ε W U U, U' \ \ V \ R c R Ο Therefore, if Η be taken on AM such that CH : Η Μ AA' : p„, Η will fall between A and M. PROPOSITIONS DETERMININO THE EVOLUTE. 173 Take two mean proportionals OiV,, CI between CA and CH*, and let P^N^ be the ordinate through iV,. Take a point L on OM (in the hyperbola) or on OM produced (in the ellipse) such that OL : LM = AA' : pa. Draw LVE, OR both parallel to the axis, and CE, HVR both perpendicular to the axis. Let the tangent at P^ meet the axis in Τ and EL in W, and let P^N^ meet EL in U^. Join 0P„ meeting the axis in K^. Let ηοΛν y be such a length that y : P^N^ = (CM : MH) . {HN^ : iV.C). (a) Suppose first that OM > y ; .•. OM'.P^N^>y:P^N^. But OM : P,i\^, = (Oil/ : ML) . (il/X : P^N^) = {OM:ML).{N^U^:P^N^), and y : PjiY, = (Ci¥ : il/^) . {ΗΛ\ : i\^,C') = (Oi¥:i/X).(iyiV^j:i\'',C); .•. N^U^:P,N^>HA\:Nfi (1), or P^N^.N^H<CN^.N^U^. Adding or subtracting the rectangle U^N^ . N^H, we have P^L\.U,V<CH.HV <LO.OR, •.• CH : HM=OL : LM. But, for a normal at Pj, we must have [from the proof of Prop. 97] P^U,.UJ=LO.OR. Therefore P^O is not a normal, and [as in the proof of Prop. 97] * For ApoUonius' method of finding two mean proportionals see the Intro- duction. 174 THE COXICS OF APOLLOXIUS. Next let Ρ be any other point than P^, and let U, N, Κ have the same relation to Ρ that U^,N^, K^ have to P,. Also, since U^N^ •Ν,Ρ,> HN^ : N<0 by (1) above, let w, be taken on i/',iV, such that η^Λ\: i\\P^ = HN, : Nfi (2), and draw wuu^v parallel to WUU^V. Now CN^ . CT = CA\ so that GN^ : CA = CA : CT ; .•. CT is a third proportional to CiV,, CA. But CX^ is a third proportional to CH, CI, CA\ :CA = CI : CN^ = CH : CI; .•. CH : CN^ = CN^ : CT = CH-^ CN^ : 6'iY, ~ CT = HN^ : N^T. CH : CN^ = P^u^ : P^iV^, since n^N^ : i\'',P, = HN^ : iV^C, from (2) above ; and And thus ΙϊΑΝ<ΑΝ^, wu < u^v, and WjW : uw > u^u : u^v, whence u^w : uw > nv : n^v. .'. P,«, : Qu > uv : u^v (where PiV meets Ρ,Γ in Q); thus PjWj . WjV > Qi< . riv > Pu . uv, afortioH. But, since HN,'.Nfi=u^N^:P^N^, P,N^.N,H = CN^.N^u^, and, adding or subtracting the rectangle i<,iV, . iV, ff, ΗΛ\ : λ\Τ = P^u^ : P^N^ = u^w .A\T; u^w = HN^ = u^v. If AN>AN^, imi^^> uv ; .". uu^ : uv > uu^ : wu^, whence w?i, : vu > wiL : wu^ > Qi(' ■ Λ". ; thus P,u^.u^v>Qu.uv > Pu . uv, a fortiori, and the proof proceeds as in the first column, leading to the same result, PU.UV<LO.OR, PROPOSITIONS DETKHMININO THE Ε\'01ΓΤΕ. 175 P^u^.u^v = CH.Hv; .•. CH.Hv>Pu.uv, and, adding or subtracting the rectangle uU. UV, PU.UV<CH.Hv + uU.UV for the hyperbola, or PU.UV<CH.Hv-uU.UV for the ellipse, .•. in either case, a fortiori, PU.UV<CH.HV, or PU.UVkLO.OR Therefore, as in the proof of Prop. 97, PO is not a normal, hx\tNK<NCT. (b) Next suppose OM = y, so that Oil/: P,JS\ = y : P,^\, and Λνβ obtain in this case υ^Ν^:Ν^Ρ^ = ΗΛ\ :Λ\0: .•. CN^.N^U^ = P^A\.N^H. Adding or subtracting U^N^.N^H, Ave have P^U^. U,V=CH.HV=LO.OR, and this [Prop. 97] is the property of the normal at P,. Therefore one normal can be drawn from 0. If Ρ be any other point on the curve, it will be shown as before that U^W= U^V, because in this case the lines WV, wv coincide ; also UU, : UW> UU, : U,V in the case where UW< U^V, and UU^ : UV> UU, -.U^W in the case where U,W > UV, whence, exactly as before, we derive that P,U,.U,V>QU.UV >PU. UV, afortion, and thence that PU . UV<LO.OR Therefore PO is not a normal, and NK < -tYG. 176 THE COXICS OF APOLLONIUS. (c) Lastly, if OM < y, wc shall have in this case G l'.•' A <9' ρ / / c s \ \ " \ „z '' A Ν Ν, Κ^ν^ :-K /\ 7^ υ υ, U' ^^^v •0 w «a PROPOSITIONS DETERMINING THE EVOLUTK. 177 and we shall derive LO.uR<PJJ^.UJ. Let S be taken on 1\λ\ such that LO .OR = SU^. i7, V, and through ά' describe a hyperbola whose asymptotes are VW and F// produced. This hyperbola will therefore meet the conic in two points P, P', and by the property of the hyperbola PU. UV=P'U'. U'V=SU^.UJ^LU.OR, so that PO, P'O are both normals. The rest of the proposition follows at once from Prop. 97. [It is clear that in case (6) is the point of intersection of two consecutive normals, or the centre of the circle of curvature at P. To find the Cartesian equation of the evolute we have X = CM, I CHa^ GH _αλ (1). Also _j^_C^HN^ GN ^ Ρ N"^ and ^ +^'';• =1 (3), where the upper sign refers to the hyperbola. And, lastly, a : GN, = GN^: GI =GI :CH (4). From (4) GN;' = a.GI, , ,,„ a.GH and 6'iV, = ^ ; .•. GA\' = a\CH (.5). ΝοΛΥ, from (2), jl_^GAI^ HN, P^N^ MR- Nfi ft* + Λ» Ρ iV ■' = ^.^ .by(3). H. C. 1 2 178 THE COXWS OF APOLLONIUS, 6> Thus P,i\7=-^ 6' whence P,N; = b\l•., ^ ^^,) (6). ax But, from (1), CH = , , ,« . ^ a ± b Therefore, by (5). CN^' = -^^—yt , whence C Ν ;' = a' . [-^^^ (7). Thus, from (6) and (7), by the aid of (3), ax \i ( by y^ a*±6V W±bV ' {ax)i + {byf = {a' ± 6^)1] Proposition lOO. [V. 53, 54.] If be a point on the minor axis of an ellipse, then (a) if OB : BG <^ A A' : pa, and Ρ be any point on either of the quadrants BA, BA' except the point B, and if OP meet the major axis in K, PO cannot be a normal, but NK < NG ; (6) if OB : BC < A A' : pa, one normal only besides OB can be drawn to either of the tivo quadrants as OP, and, if P' be any other point, N'K' is less or greater than N'G' according as P' is further from, or nearer to, the minor axis than P. [This proposition follows at once as a particular case of the preceding, but Apollonius proves it separately thus.] (a) We have OB : BC < On: jiC ; .•. On : nC, or CN:NK> AA' : pa, whence CN : NK > ON : NG, and NK<NG. PROPOSITIONS DETERMINING THE KVOLUTE. (b) Suppose now that 0'B:BC<AA':pa. Take a point η on O'B such that O'n : nC = AA' -.pa- 179 ρ y^^ η X ( ^ \ Μ \k \G C j o' ^y ο Therefore CN : NK, = AA' : pa, where Ν is the foot of the ordinate of P, the point in which nP draAvn parallel to the major axis meets the ellipse, and K^ is the point in which O'P meets the major axis ; .• . iVifj = NG, and PO' is a normal. PO', BO' are then two normals through 0', and the rest of the proposition follows from Prop. 97. 12—2 CONSTRUCTION OF NORMALS. Proposition lOl. [V. 5o, 5G, 57.] If is any point beloiv the axis A A' of an ellipse, and AM > AC {where Μ is the foot of the perpendicular from on the aids), then one normal to the ellipse can always be drawn tlirough cutting the a:vis between A and C, but never more than one such normal. Produce OM to L and CM to Η so that OL : LM= CH : HM= AA' : p^, and draw LI, IH parallel and perpendicular to the axis respectively. Then with IL, I Η as asymptotes describe a [rectangular] hyperbola passing through 0. CONSTRUCTION OF NORMALS. 181 This will meet the ellipse in some point P,. For, drawing AD, the tangent at A, to meet IL produced in I), we have AH:HM>CH:HM > AA' : j)a > OL : LM: .•. AH.LM>OL.HM, or AD.DI>UL.LL Thus, from the property of the hyperbola, it must meet xiD between A and D, and therefore must meet the ellipse in some point P,. Produce OP^ both ways to meet the asymptotes in R, R', and draw R'E perpendicular to the axis. Therefore OR=P^R', and consequently EN^ = iMH. Now AA' :pa = OL:LM = ME : EK^, by similar triangles. Also AA':pa = CH:HM; .•. AA' : Pa = -1/^ - CH : EK^ - MH since EN^ = MH. Therefore N^K^ = N^G^, and P^O is a normal. Let Ρ be any other point such that OP meets AC in K. Produce BC to meet OP, in F, and join FP, meeting the axis in K'. Then, since two normals [at P,, B] meet in F, FP is not a normal, but NK' > NG. Therefore, α fortiori, NK > NG. And, if Ρ is between A and P„ Ν Κ < NG. [Prop. 97, Cor. 1.] 182 THE comes OF APOLLONIUS. Proposition 102. [V. 58, 59, GO, 61.] If be any point outside a conic, but not on the ct-xis iuhose extremity is A, we can draw a normal to the curve through 0. For the parabola we have only to measure MH in the direction of the axis produced outside the curve, and of length equal to ^ , to draw HR perpendicular to the axis on the same side as 0, and, with HR, HA as asymptotes, to describe a [rectangular] hyperbola through 0. This will meet the curve in a point P, and, if OP be joined and produced to meet the axis in Κ and HR in R, we have at once Η Μ = NK. Therefore and PK is a normal. NK^P^. In the hyperbola or ellipse take Η on CM or on CM ])Γθ(1π(•.•(1, and L on OM or OM produced, so that 67/ :HM=OL:LM=AA' Pa- CONSTRUCTION OF NORMALS. 183 Then draw HIR perpendicular to the axis, and ILW through L parallel to the axis. Β / ? > \ ^ A Μ V νΛ c ) \ . I J R' (1) If Μ falls on the side of C towards A, draw with asymptotes IR, IL, and through 0, a [rectangular] hyperbola cutting the curve in P. (2) If Μ falls on the side of C further from A in the hyperbola, draw a [rectangular] hyperbola with IH, IR' as asymptotes and through C, the centre, cutting the curve in P. 184 THE COXICS OK APOLLONIUS. Then OP will be a normal. For we have (1) Μ Κ : HN = MK : LR', since OR = PR', and therefore IL = UR'. Therefore MK : Η Ν = MO : OL, by similar triangles, = MC : CH, •.• CH : HM = OL : LM. Therefore, alternately, MK:MC=NH:HG (A). In case (2) OL : LM = CH : HM, or OL.LI=GH.HI, [so that 0, C are on opposite branches of the same rectangular hyperbola]. Therefore PU : OL = LI : lU, or, by similar triangles, UR'.R'L^LI-.IU, whence R'L = IU=HN] .•. MK.HN=MK'.R'L = MO : OL = MG : GH, and MK : MG = NH : iTC, as before (A). Thus, in either case, we derive GK : GM=GN:GH, and hence, alternately, GN.GK = GH:GM, so that GN:NK=GH: HM = AA':pa\ .•. NK = NG, and oy is the normal at P. I CONSTRUCTION OF NORMALS. 185 (3) For the hyperbola, in the particuhir case where Μ coincides with C, or is on the conjugate axis, wc need only divide OC in L, so that OL : LC=AA':pa, and then diaw LP parallel to AA' to meet the hyperbola in 1\ Ρ is then the foot of the normal through 0, for AA' ■.pa=uL: LC = OP : Ρ Κ = CN.NK, and NK^NCr. [The particular case is that in which the hyperbola used in the construction reduces to two straight lines.] Proposition 103. [V. 62, 63.] If ϋ he an internal point, we can draw through (J a normal to the conic. 186 THE COXICS OF APOLLONIUS. The construction and proof proceed as in the preceding proposition, mutatis mutandis. The case of the parabola is obvious ; and for the hi/perhola or ellipse MK.HN=OM: OL = CM : CH. .•. CM : CH = CM ± MK : CH ± HN = CK:CN• .•. NK:CN = HM.CH = 2^a :AA'\ .•. NK=NG, and PO is a normal. OTHER PROPOSITIONS RESPECTING MAXIMA AND MINIxMA. Proposition 104. [V. 64, (J5, 66, 67.] If be a jjoint below the axis of any conic such that either no normal, or only one normal, can be drawn to the curve through which cuts the aa-is {betiueen A and C in the case of the ellipse), then OA is the least of the lines OP cutting the axis, and that which is nearer to OA is less than that which is more remote. If OM be perpendicular to the axis, we must have AM>^, and also OM must be either greater than or equal to y, where (a) in the case of the parabola ij.P^N^ = N^H:HM: (6) in the case of the hyperbola or ellipse with the notation of Prop. 99. In the case where OM > y, we have proved in Prop. 99 for all three curves that, for any straight line OP drawn from to the curve and cutting the axis in K, NK< NG ; but, in the case where OM = y, Ν Κ < NG for any point Ρ between A and P, except P, itself, for which N^K^ = N^G^. 188 THE CONICS OF APOLLONIUS. Also for any point Ρ more remote from A than P^ it is still true that Ν Κ < NG. I. Consider now the ease of any of the three conies where, for all points P, NK < NG. Let Ρ be any point other than A. Draw the tangents A F, PT. Then the angle OA Υ is obtuse. Therefore the per- pendicular at A to AO, as AL, falls within the curve. Also, since Ν Κ < NG, and PG is perpendicular to PT, the angle OPT is acute. (1) Suppose, if possible, UP= OA. With OP as radius and as centre describe a circle. Since the angle OPT is acute, this circle will cut the tangent PT, but AL will lie wholly without it. It follows that the circle must cut the conic in some intermediate point as R. li RU be the tangent to the conic at R, the angle ORU is acute. Therefore RU must meet the circle. But it falls wholly outside it : which is absurd. Therefore OP is not equal to OA. (2) Suppose, if possible, OP < OA. OTHER PROPOSITIONS RESPECTING MAXIMA AND MINIMA. 189 In this case the circle drawn with Ο as centre and UP as radius must cut AM in some point, D. And an absurdity is proved in the same manner as before. Therefore OP is neither etjual to (J A nor loss than OA, i.e. ()A < OP. It remains to be proved that, if P' be a point beyond P, OP < OP'. If the tangent TP be produced to T', the angle OPT' is obtuse because the angle OPT is acute. Therefore the perpen- dicular from Ρ to OP, viz. PE, ialls within the curve, and the same proof as was used for A, Ρ will apply to P, P'. Therefore OA < OP, OP < OP', &c. II. Where only one normal, 0P^, cutting the axis can be drawn from 0, the above proof applies to all points Ρ between A and P, (excluding P, itself) and also applies to the comparison between tAvo points Ρ each of which is more remote from A than P. 190 THE COXK'S OF APOLLONIUS. It only remains therefore to prove that (a) OP^ > any straight line OP between 0Λ and OP^, Φ) OP^ < any straight line OP' beyond OP^. (a) Suppose first, if possible, that OP = OP^, and let Q be any point between them, so that, by the preceding proof, OQ > OP. Measure along OQ a length Oq such that Oq is greater than OP, and less than OQ. With as centre and Oq as radius describe a circle meeting OP^ produced in p^. This circle must then meet the conic in an intermediate point R. Thus, by the preceding proof, OQ is less than OR, and there- fore is less than Oq : which is absurd. Therefore OP is not equal to OP^. Again suppose, if possible, that OP > OP^. Then, by taking on OP, a length 0;j, greater than OP^ and less than OP, an absurdity is proved in the same manner. Therefore, since OP is neither equal to nor gi-eater than OP^, OP<OP^. (b) If OP' lies more remote from 0Λ than 07-*,, an exactly similar proof will show that OP^ < OP'. Thus the proposition is completely established. Proposition 105. (Lemma.) [V. 68, 69, 70, 71.] If two tangents at points Q, Q' on one side of the aads of a conic meet in T, and if Q be nearer to the axis than Q', then TQ < TQ'. The propcjsition is proved at once for the parabola and hyperbola and for the case where Q, Q' are on one quadrant of an ellipse: for the angle TVQ' is greater than the angle TVQ, and QV=Vq. OTHER PROPOSITIONS RESPECTING MAXIMA AND MINIMA. 191 Therefore the base TQ is less than the base TQ'. In the case where Q, Q' are on different quadrants of an ellipse, produce the ordinate Q'N' to meet the ellipse again in q. Join q'C and produce it to meet the ellipse in R. Then Q'N' = N'q', and q'G= CR, so that Q'R is parallel to the axis. Let RM be the ordinate of R. NOAV .•. [Prop. 86, Cor.] and, as before. RM>QN; CQ > CR, >0Q'; .•. zGVQ>zGVQ' TQ<TQ'. Proposition 106. [V. 72.] If from a point below the axis of a jxirabola or hyperbola it is possible to draw two normals OP^, OP^ cutting the axis (P, being nearer to the vertex A than P^), and if further Ρ be any othei• point on the carve and UP be joined, then 192 THE COXICS OF APOLLONIUS. (1) if Ρ lies behueen A and F^, OP^ is the greatest of all the lines OP, and that which is nearer to OP^ on each side is greater than that which is more remote; (2) if Ρ lies between P, and P^, or beyond P.^, OP^ is the least of all the lines OP, and the nearer to OP^ is less than the more remote. By Prop. 99, if Ρ is between Λ and P,, OP is not a normal, but NK < NG. Therefore, by the same proof as that employed in Prop. 104, we find that OP increases continually as Ρ moves from A towards P,. We have therefore to prove that OP diminishes continually as Ρ moves from P, to Pj. Let Ρ be any point between P, and Pj, and let the tangents at Pj, Ρ meet in T. Join OT. Then, by Prop. 105, ΓΡ, < TP. Also ΓΡ,» + OP^' > TP' + 0P\ since AK > AG, and consequently the angle OPT is obtuse. Therefore OP < OP^. Similarly it can be proved that, if P' is a point between Ρ andP,, OP'kOP. That OP increa.ses continually as Ρ moves from P, further away from A and P^ is proved by the method of Prop. 104. Thus the proposition is established. OTHER MAXIMA AND ΜΙΧΙΜΛ. 193 Proposition 1 7 . [V. 73.] If be a point below the major axis of an ellipse sucJi that it is possible to draio through one normal only to the ivhole of the semi-ellipse ABA', then, ifOP^ be that normal and P, is on the quadrant AB, OP^ nill he the greatest of all the straight lines drawn from to the semi-ellipse, and that which is nearer to OP^ luill be greater than that which is more remote. Also OA' will be the least of all the straight lines drawn from to the semi-ellipse. It follows from Props. 99 and 101 thcat, if OM be per- pendicular to the axis, Μ must lie between C and A', and that OAI must be greater than the length y determined as in Prop. 99. Thus for all points Ρ between A' and B, since Κ is nearer to A' than G is, it is proved by the method of Prop. 104• that OA' is the least of all such lines OP, and OP increases con- tinually as Ρ passes from A' to B. For any point P' between Β and P, we use the method of Prop. 106, drawing the tangents at P' and B, meeting in T. u. c. 13 194 THE COXICS OF APOLLONIUS. Thus we derive at once that OB < 0P\ and similarly that OP' increases continually as P' passes from Β to P^. For the part of the curve between P, and A we employ the method of reductio ad absurdum used in the second part of Prop. 104. Proposition 108. [V. 74.] If be a point below the major ao^is of an ellipse such that two normals only can be draiun through it to the whole semi- ellipse ABA', then that normal, OP^, which cuts the minor a^is is the greatest of all straight lines from to the semi-ellipse, and that which is nearer to it is greater than that which is more remote. Also OA, joining to the nearer vertex A, is the least of all such straight lines. It follows from Prop. 99 that, if be nearer to A than to A', then P,, the point at which is the centre of curvature, is on the quadrant AB, and that OP^ is one of the only two possible normals, Avhile P^, the extremity of the other, is on the quadrant Β A' ; also 0M=y determined as in Prop. 99. In this case, since only one normal can be drawn to the quadrant AB, we prove that OP increa.ses as Ρ moves from A to P, by the method of Prop. 104, as also that OP increases as Ρ moves from P, to B. That OP increases as Ρ moves from Β to P^, and diminishes as it passes from P^ to A', is established by the method employed in the last proposition. OTHER MAXIMA AND MINIMA. 195 Proposition 109. [V. 75, 76, 77.] //' he a point below the major axis of an ellipse such that three normals can be draxun to the semi-ellipse ABA' at points Pj, Pj, P3, tuhere P,, P^ are on the quadrant AB and P^ on the quadrant BA', then (if P^ be nearest to the vertex A), (1) OP^is the greatest of all lines drawn from to points on the semi-ellipse between A' and P^, and the nearer to OP^ on either side is greater than the more remote ; (2) OP^ is the greatest of all lines from to points on the semi-ellipse from A to P^, and the nearer to OP^ on either side is greater than the more remote, (3) of the two majdma, OP3 > OP^. Part (2) of this proposition is established by the method of Prop. 106. p^ Part (1) is proved by the method of Prop. 107. It remains to prove (3). a| We have GN^ •.N^G^ = A A' : p^ = CN^ : Νβ^ ; < MN^ : Νβ^, a fortiori, whence MG, : Ν β, < MG, : Nfi, ; and, by similar triangles, OM.P^N^<OM:P,N^, or P,N^ > P,N,. If then Pjj^ be parallel to the axis, meeting the curve in jt), , we have at once, on producing OM to R, P,R>PA so that Op, > OP, ; .•. a fortiori 0P^> OP,. 13—2 196 THE COXICS OF APOLLONIUS. As particular cases of the foregoing propositions we have (1) If be on the minor axis, and no normal except OB can be drawn to the ellipse, OB is greater than any other straight line ft-om to the curve, and the nearer to it is greater than the more remote. (2) If be on the minor axis, and one normal (besides OB) can be drawn to either quadrant as OP,, then OP^ is the greatest of all straight lines from to the curve, and the nearer to it is greater than the more remote. EQUAL AND SIMILAR CONICS. Definitions. 1. Conic sections are said to be equal Avhen one can be applied to the other in such a way that they everywhere coincide and nowhere cut one another. When this is not the case they are unequal. 2. Conies are said to be similar if, the same number of ordinates being drawn to the axis at proportional distances from the vertex, all the ordinates are respectively proportional to the corresponding abscissae. Otherwise they are dissimilar. 3. The straight line subtending a segment of a circle or a conic is called the base of the segment. 4. The diameter of the segment is the straight line which bisects all chords in it parallel to the base, and the point where the diameter meets the segment is the vertex of the segment. 5. Equal segments are such that one can be applied to the other in such a way that they everywhere coincide and nowhere cut one another. Otherwise they are unequal. 6. Segments arc similar in which the angles between the respective bases and diameters are equal, and in which, parallels to the base being drawn from points on each segment to meet the diameter at points proportionally distant from the vertex, each parallel is respectively proportional to the corresponding abscissa in each. 198 THE COXICS OF APOLLONIUS. Proposition llO. [VI. 1, 2.] (1) In two parabolas, if the ordinates to a diameter in each are inclined to the respective diameters at equal angles, and if the corresponding parameters are equal, the ttuo parabolas are equal. (2) If the ordinates to a diameter in each of two hyperbolas or two ellipses are equally inclined to the respective diameters, and if the diameters as well as the corresponding parameters are equal respectively, the two conies are equal, and conversely. This proposition is at once established by means of the fundamental properties ( 1 ) QV' = PL.PV for the parabola, and (2) QV* = PV.VR for the hyperbola or ellipse proved in Props. 1 — 3. Proposition 111. [VI. 3.] Since an ellipse is limited, tvhile a parabola and a hyperbola proceed to infinity, an ellipse cannot be equal to either of the other curves. Also a parabola cannot be equal to a hyperbola. For, if a parabola be equal to a hyperbola, they can be applied to one another so as to coincide throughout. If then eijual abscissae AN, AN' be taken along the axes in each we have for the parabola AN : AN' = PN' : P'N'\ Therefore the same holds for the hyperbola : which is im- possible, because PN' : P'N" = AN.A'N : AN' . A'N'. Therefore a parabola and hyperbola cannot be equal. [Here follow six easy propositions, chiefly depending upon the symmetrical form of a conic, which need not be re- produced.] EQUAL AND SIMILAR CONICS. 199 Proposition 112. [VI. 11, 12, 13.] (1) All parabolas are similar. (2) Hyperbolas, or ellipses, are similar to one another when the "figure" on a diameter of one is similar to the "figure" on a diameter of the other and the ordinates to the diameters in each make equal angles ivith the diameters respectively. (1) The result is derived at once from the property FN'=Pa.AK (2) Suppose the diameters to be axes in the first place (conjugate axes for hyperbolas, and both major or both minor axes for ellipses) so that the ordinates are at right angles to the diameters in both. Then the ratio pa : AA' is the same in both curves. There- fore, using capital letters for one conic and small letters for the other, and making AN : an equal to AA' : aa', we have at the same time PN^ : AN. Ν A' =pn' : an.na'. But AN. Ν A' : AN^ = an . na' : α/^^ because A'N : AN= a'n : an ; .•. PN'':AN'=pn':an\ or PN : AN = pn : an, and the condition of similarity is satisfied (Def. 2). Again, let ΡΡ',ρρ be diameters in two hyperbolas or two ellipses, such that the corresponding ordinates make equal angles with the diameters, and the ratios of each diameter to its parameter are equal. Draw tangents at P, ρ meeting the axes in T, t respectively. Then the angles CPT, cpt are equal. Draw AH, ah perpen- dicular to the axes and meeting CP, cp in H, h ; and on GH, ch as diameters describe circles, Avhich therefore pass respectively through A, a. Draw QAR, qar through A, a parallel respec- tively to the tangents at P, ρ and meeting the circles just described in R, r. 200 THE COXICS OF APOLLONIUS. Let V, V be the middle points of AQ, aq, so that V, ν lie on CP, cp respectively. Then, since the "figures" on PP' , ]ψ' are similar, AV':CV.VH= av' : cv . vh, [ Prop. 1 4] or AV':AV.VR = av':av.vr, whence AV : VR = av : vr {a), and, since the angle A VC is etpial to the angle avc, it follows that the angles at C, c arc etjual. EQUAL AND SIMILAR CONICS. 201 [For, if K, k be the centres uf the circles, and /, i the middle points 0Ϊ AR, ar, we derive from (a) VA : AI = va : ai ; and, since ZKVI= Ζ kvi, the triangles KVI, kvi are similar. Therefore, since VI, vi are divided at -i4, α in the same ratio the triangles KVA, kva are similar; .•. ZAKV= Zakv: 202 THE COXICS OF APOLLONIUS. hence the halves of these angles, or of their supplements, are equal, or Ζ KG A = Ζ kca.] Therefore, since the angles at F, j) are also equal, the triangles CFT, cpt are similar. Draw PiV,p/i perpendicular to the axes, and it will follow that FN':CN.NT = 2^n'-cn.nt, whence the ratio of ΛΑ' to its parameter and that of «a' to its parameter are equal. [Prop. 14] Therefore (by the previous case) the conies are similar. Proposition 113. [VI. 14, 15.] A parabola is neither similar to a hyperbola nor to an ellipse ; and a hyperbola is not similar to an ellipse. [Proved by reductio ad absurdum from the ordinate pro- perties.] Proposition 114. [VI. 17, 18.] (1) If FT,pt be tangents to tivo similar conies meeting the axes in T, t respectively and making equal angles with them; if, further, FV, ρυ be measured along the diameters through F, ρ so that FV:FT = pv:pt, and if QQ', qq be the chords through V, ν parallel to FT, pt respectively: then the segments QFQ', gpq' are similar and similarly situated. (2) And, conversely, if the segments are similar and simiUirly situated, FV: FT = pv :pt, and the tangents are equally inclined to the axes. EQUAL AND SIMILAR CONICS. 203 I. Let the conies be parahokis. Draw the tangents at A, a meeting the diameters through P, ρ in H, It, and let PL, pi be such lengths that PL : 2PT = OP : PH\ and pi : 2pt = op -.pit, ) where 0, ο are the points of intersection of AH, PT and ah, pt. Therefore PL, pi are the parameters of the ordinates to the diameters PV, pv. [Prop. 22] Hence QV' = PL.PV, qv^ = pi . pv. (1) ΝοΛν, since zPTA=Zpta, Z0PH= Zoph, and the triangles ΟΡΗ, oph are similar. Therefore OP : Ρ Η = op : ph , so that PL : PT = pi : pt. But, by h}^othesis, PV:PT = pv:pt\ .•. PL:PV = pl:pv, and, since QV is a mean proportional between PV, PL, and qv between pv, pi, QV:PV=qv .pv. 204 THE COXICS OF APOLLONIUS. Similarly, if V, v' be points on PV, pv such that PV: PV'=2)v :pv', and therefore PL : PV =pl : pv', it follows that the ordinates passing through V, v' are in the same ratio to their respective abscissae. Therefore the segments are similar. (Def. 6.) (2) If the segments are similar and similarly situated, Λνβ have to prove that ΔΡΤΑ = Zpta, and PV : PT = pv : 2)t. Now the tangents at P, ρ are parallel to QQ', qq' respec- tively, and the angles at V, ν are equal. Therefore the angles PTA,pta are equal. Also, by similar segments, QV: PV=qv : pv, while PL : QV = QV : PV, and pi : qv = qv :pv\ .•. PL:PV=pl:pv. But PL : 2PT = OP : PH) pi : "Ipt = op : ph j ' and UP : PH= op : ph, by similar triangles. Therefore PV : PT = pv : pt. II. If the curves be hyperbolas or ellipses, suppose a similar construction made, and let the ordinates PN, pn be drawn to the major or conjugate axes. We can use the figures of Prop. 112, only remembering that the chords arc here QQ', qq', and do not pass through A, a. (1) Since the conies are similar, the ratio of the axis to its parameter is the same for both. EQUAL AND SIMILAR CONICS. 205 Therefore FX' : CN . NT = pn' : en . nt. [ Prop. 1 4] Also the angles PTN, ptn are diual, therefore PN : NT = pn : nt. Hence PN : CN =pn : en, and ZPCN= Ζ pen. Therefore also ζ CPT= Ζ cpt It follows that the triangles ΟΡΗ, oph are similar. Therefore OP : PH = op : ph. But OP : PH = PL : 2PT\ op : ph=pl •.2pt j ' whence PL : PT = pl : pt. Also, by similar triangles, PT :GP=pt:ep; .•. PL:CP=pl:cp, or PL: PP'=pl.pp' (A). Therefore the "figures" on the diameters PP', pp' are similar. Again, we made PV : PT =pv : pt, so that PL: PV = pl : pv (B). We derive, by the method employed in Prop. 112, that QV:PV=qv:pv, and that, \{ ΡΥ,ρν be proportionally divided in the points V, v, the ordinates through these points are in the same ratios. Also the angles at V, ν are equal. Therefore the segments are similar, (2) If the segments are similar, the ordinates are in the ratio of their abscissae, and we have QV:PV=qv :pv PV:PV' = pv :pv' oV':Q'V'--=pv' :q'v 206 THE COXICS OF APOLLONIUS. Then QV: Q' V" = qv' : q'v" ; .•. PV.VP'-.PV. V'P'=pv.vp''.pv.v'p, and PV: PV =pv : pv', so that P'V : P'V =p'v : p'v'. From these equations it follows that py : VV'=pv' :vv') and P'V : FF' = jjV : vy'j ' whence P'V : Ρ V = p'v' : pv ; .•. P' V . VP -.PV'^ p'v' . v'p : pv'*. But PV':Q'V'=pv":q'v'*; .•. P'F'. F'P : Q'V"=p'v'.v'p : q'v'^. But these ratios are those of PP', pp' to their respective parameters. Therefore the "figures" on PP', pp are similar; and, since the angles at F, ν are equal, the conies are similar. Again, since the conies are similar, the " figures " on the axes are similar. Therefore PN"" : C'iV . NT = pn' : C7i . nt, and the angles at N, η are right, while the angle CPT is equal to the angle cpt. Therefore the triangles CPT, cpt are similar, and the angle CTP is equal to the angle ctp. Now, since PV. VP' : QV^ = pv . vp' : qv^, and QV:PV' = qv':pv'\ it follows that PV : P'V ==pv : p'v, whence PP' : PV = pp' : pv. But, by the similar triangles CPT, cpt, CP : PT = cp : pt, or PP' :PT = pp' :pt; .•. PV: PT = pv:pt, and the proposition is proved. EQUAL AND SIMILAR CONICS. 207 Proposition 115. [VI. 21, 22.] If two ordinates he drawn to the axes of two parabolas, or the major or conjugate axes of two similar ellipses or two similar hyperbolas, as PN, P'N' andpn, p'n, such that the ratios AN : on and AN' : an' are each equal to the ratio of the respective latera recta, then the segments PP', pp will he similar ; also PP' will not he similar to any segment in the other conic which is cut off by ttvo ordinates other than pn, p'n, and vice versa. [The method of proof adopted follows the line.s of the previous propositions, and accordingly it is unnecessary to reproduce it] Proposition 116. [VI. 26, 27.] If any cone be cut by two parallel planes making hyperbolic or elliptic sections, the sections will be similar but not equal. On referring to the figures of Props. 2 and 3, it will be seen at once that, if another plane parallel to the plane of section be drawn, it will cut the plane of the axial triangle in a straight line p'pm parallel to P'PM and the base in a line dme parallel to DME; also p'pm will be the diameter of the resulting hyperbola or ellipse, and the ordinates to it will be parallel to dme, i.e. to DME. Therefore the ordinates to the diameters are equally inclined to those diameters in both curves. Also, if PL, pi are the corresponding parameters, PL : PP' = BF. FC -.AF'^pl: pp. '^ crrvr. 208 THE COXICS OF APOLLONIUS. Hence the rectangles PL . PP' and i)l .pp are similar. It follows that the conies are similar. [Prop. 112] And they cannot be equal, since PL . PP' cannot be equal to 2)1. pp. [Cf. Prop. 110(2)] [A similai• proposition holds for the parabola, since, by Prop. 1, PL : ΡΛ is a constant ratio. Therefore two parallel parabolic sections have different parameters.] PROBLEMS. Proposition 117. [VI. 28.] In a given right cone to find a parabolic section equal to a given parabola. Let the given parabola be that of which am is the a.xis and al the latus rectum. Let the given right cone be OBO, where is the apex and BC the circular base, and let OBC be a triangle through the axis meeting the base in BC. Measure 0Λ along OB such that al : OA = B(f : BO . 0(1 H. C. 14 210 THE COXIt'S OF APOLLONIUS. DraAV AM parallel to OC meeting BG in M, and through AM draw a plane at right angles to the plane OBC and cutting the circuhvr base in DME. Thi'u T)E is perpendicular to AM, and the section DAE is a parabola whose axis is AM. Also [Prop. 1], \ϊ AL is the latus rectum, AL:AO = BG' .BO. 00, whence AL = aI, and the parabola is equal to the given one [Prop. 110]. No other parabola with vertex on OB can be found which is equal to the given parabola except DAE. For, if another such parabola were possible, its plane must be perpendicular to the plane OBC and its axis must be parallel to 00. If A' were the supposed vertex and A'L' the latus rectum, we should have A'L' : A'O = BG^ •. BO . 00 = AL : AO. Thus, if A' does not coincide with A, A'L' cannot be equal to AL or al, and the parabola cannot be equal to the given one. Proposition 118. [VI. 29.] Ln a given right cone to find a section equal to a given hyperbola. {A necessary condition of possibility is that the 7'atio of the square on the axis of the cone to the square on the radius of the base must not be greater titan the ratio of the transverse a.vis of the given hyperbola to its parameter.) Let the given hyperbola be that of which aa', al are the transverse axis and parameter respectively. I. Suppose 07" : BP < aa' : al, Λvhere I is the centre of the base of the given cone. Let a circle be circumscribed about the axial triangle OBC, and produce 01 to meet the circle again in D. PROBLEMS. Then OI:TD= OP : BI that 01 ■.ID< aa' : al. 211 Take Ε on ID such that 01 : IE = aa' : al, and through Ε draw the chord QQ' parallel to BC. Suppose now that ΛΑ',Λ^Λ^' are placed in the angle formed by 00 and BO produced, such that AA' = A^A^' = aa', and AA', -4, J.,' are respectively parallel to OQ, OQ', meeting BG in M, M'. Through A' AM, A^A^M' draw planes perpendicular to the plane of the triangle OBG making hyperbolic sections, of which A' AM, A^A^M' will therefore be the transverse axes. Suppose OQ, OQ' to meet BC in F, F'. Then aa' : al =01 .IE = OF:FQ or OF' : F'Q' = OF^.OF.FQ or OF"" : OF' . F'Q' = or : BF. FC or OF" : BF' . i"C = .4yl':yl/v..r .I,yl,': Λ J.,. 14—2 212 THE COXICS OF APOLLONIUS, where AL, AJj^ arc the parameters of AA', A^A^ in the sections respectively. It follows, since A A' = A^A' = aa', that AL = AJ.^=al. Hence the two hyperbolic sections are each equal to the given hyi)crbola. There are no other equal sections having their vertices on 00. For ( 1 ), if such a section were possible and OH were parallel to the axis of such a section, OH could not be coincident either Avith OQ or OQ'. This is proved after the manner of the preceding proposition for the parabola. If then (2) OH meet BO in H, QQ in R, and the circle again in K, we should have, if the section w^ere possible, aa' :al=OH^'.BH.HC = 0H': OH.HK = OH.HK; which is impossible, since aa':al=OI ■.IE=OH:HR. II. If or : ΒΓ = aa' : al, we shall have 01 : ID = aa' : al, and OQ, OQ will both coincide with OD. In this case there will be only one section equal to the given hyperbola whose vertex is on OC, and the axis of this section will be perpendicular to BC. III. If OP : BP > aa' : al, no section can be found in the right cone which is equal to the given hyperbola. For, if possible, let there be such a section, and let ON be drawn parallel to its axis meeting BG in N. Thon we must have aa' : al = ON'' : BN . NO, so that OP :BI.IC> ON^ : BN. NO. But ON'>OP, while nr. Τ0>ΒΝ. NC•. which is absurd. I'RORLEMS. •2\:\ Proposition 119. LVL 80.] In a given right cone to find a section equal to a given ellipse. In this ciise we describe the circle about OBG and suppose F, F' taken on BO produced in both directions such that, if OF, OF' meet the circle in Q, Q', OF:FQ=OF':F'Q' = , at. Then we place straight lines ΑΛ', ^1,/!,' in the angle BOG so that they are each equal to aa\ while ^1.1' is parallel to OQ and A^A; to OQ. Next suppose planes drawn through A A', A^A^' each perpendicular to the plane of OBC, and these planes determine two sections each of which is equal to the given ellipse. The proof follows the method of the preceding proposition. 214 THE coyics of apollonius. Proposition 120. [VI. 31.] To find a rir/ht cone similar to a given one and containing a given parabola as a section of it. Let OBC be an axial section of the given right cone, and let the given parabola be that of which AN is the axis and AL the latus rectum. Erect a plane passing through AN and perpendicular to the plane of the parabola, and in this plane make the angle NAM equal to the angle OBC. Let AM be taken of such a length that AL : AM= EG : BO, and on AM as base, in the plane MAN, describe the triangle Ε AM similar to the triangle OBC. Then suppose a cone described with vertex Ε and base the circle on AM as diameter in a plane perpendicular to the plane Ε AM. The cone Ε AM will be the cone required. For δΜΑΝ = δΟΒΟ = δΕΑΜ = δΕΜΑ', therefore EM is parallel to AN, the axis of the parabola. Thus the plane of the given parabola cuts the cone in a section which is also a parabola. Now AL:AM = BG:BO = AM:AE, or AM' = EA.AL; .'. AM' •.AE.EM = AL.EM = AL -.EA. PROBLEMS. 21 ό Hence AL is the latus rectum of the })arabolic section ot" the cone made by the plane of the given parabohi. It is also the latus rectum of the given parabola. Therefore the given parabola is itself the parabolic section, and Ε AM is the cone required. There can be no other right cone similar to the given on•.•, having its vertex on the same side of the given parabola, and containing that parabola iis a section. For, if another such cone be possible, with vertex F, draw through the axis of this cone a plane cutting the plane of the given parabola at right angles. The planes must then intersect in AN, the axis of the parabola, and therefore F must lie in the plane of ^^lY. Again, if AF, FR are the sides of the axial triangle of the cone, FR must be parallel to liN, or to EM, and ^AFR = aBOC=aAEM, so that F must lie on ^^ or ΑΕ produced. Let AM meet FR in R. Then, if ^X' be the latus rectum of the parabolic section of the cone FAR made by the plane of the given parabola, AL' :AF = AR':AF.FR = AM':AE.EM = AL:AE. Therefore AL', AL cannot be equal; or the given parabola is not a section of the cone FA R. Proposition 121. [VI. 32.] To find a nr/ht cone similar to a given one and containing a given liyperhoki as a section of it. {If OBC be the given cone and D the centre of its base BG, and if A A', AL be the axis and parameter of tJie given hyperbola, a necessary condition of possibility is that the ratio OB' : DB'^ must not be greater than the ratio AA' : AL.) 216 THE LVyiCS OF APOLLONIUS. Let a plane be drawn through the axis of the given hyperbola and perpendicular to its plane; and on Λ'Λ, in the plane so described, describe a segment of a circle containing an /p Ε '/V Ο Μ f^j^^'^A'^^ ^^^-^a'V 4r -^ ^^■ k\ t1 1 A γ / angle equal to the exterior angle BOG at the vertex of the given cone. Complete the circle, and let EF be the diameter of it bisecting AA' at right angles in 1. Join A'E, AE, and draw AQ parallel to EF meeting A'E produced in G. Then, since EF bisects the angle A'E A, the angle EGA ] is equal to the angle Ε AG. And the angle AEG is equal 1 to the angle BOG, so that the triangles Ε AG, OBG are similar. Draw EM perpendicular to AG. Then OD'' : DB^ = EM' : MA' = I A"- : ΕΓ = FI : IE. I. Suppose that so that OB' : BB' < AA' : AL, FT: TE<AA':AL. puom.KMs. 217 Take a point Η on EI such that FI : IH = AA' : AL, and through /Γ draw the chord QQ' of the circle parallel to AA'. Join A'Q, AQ, and in the plane of the circle draw AR making with AQ an angle equal to the angle OBG. Let AR meet A'Q produced in R, and QQ' produced in N. Join FQ meeting ^^' in K. Then, since the angle QAR is equal to the angle OBC, and ^FQA = \^A'QA = \^BOC, AR'iH parallel to FQ. Also the triangle QAR is similar to the triangle OBG. Suppose a cone formed with vertex Q and base the circle described on J.ii as diameter in a plane perpendicular to that of the circle FQA. This cone will be such that the given hyperbola is a section of it. We have, by construction, AA' : AL = FI -.IH = FK : KQ, by parallels, ^FK.KQ-.KQ' = A'K.KA :KQ\ But, by the parallelogram QKAN, A'K:KQ=^QN:NR, and KA:KQ = QN : Ν A, whence A' Κ . ΚΑ : KQ' = QN' : AN . NE. It follows that AA':AL = QN':AN.NR. Therefore [Prop, 2] AL is the parameter of the hyperbolic section of the cone QAR made by the plane of the given hyperbola. The two hyperbolas accordingly have the same axis and parameter, whence they coincide [Prop. 110 (2)]; and the cone QAR has the re([uired property. 218 THE coyics of apollonius. Another such cone is found by taking the point Q' instead of Q and proceeding as before. No other right cone except these two can be found which is similar to the given one, has its apex on the same side of the plane of the given hyperbola, and contains that hyperbola as a section. For, if such a cone be possible with apex P, draw through its axis a plane cutting the plane of the given hyperbola at right angles. The plane thus described must then pass through the axis of the given hyperbola, whence Ρ must lie in the plane of the circle FQA. And, since the cone is similar to the given cone, Ρ must lie on the arc A'QA. Then, by the converse of the preceding proof, we must have (if FP meet A'A in T) AA':AL=FT:TP; .•. FT.TP = FI: IH, which is impossible. II. Suppose that OD' : ΌΒ' = AA' : AL, so that FI : IE = AA' : AL. In this case Q, Q' coalesce Avith E, and the cone with apex Ε and base the circle on AG as diameter perpendicular to the plane of FQA is the cone required. III. If UD-: DB''>AA' : AL, no right cone having the desired properties can be drawn. For, if possible, let Ρ be the apex of such a cone, and we shall have, as before, FT:TP = AA'.AL• But AA' : AL < OD' : DB', or FI : IE. Hence FT : TP < FI : IE, which is absurd. Therefore, etc. PROBLEMS. 21ί) Proposition 122. [VI. :VA.] Τυ find a right cone similar to a given one and containing a given ellipse as a section of it. As before, take a plane through ΑΛ' perpendiciUar to the plane of the given ellipse ; and in the plane so drawn describe on AA' as base a segment of a circle containing an angle equal to the angle BOC, the vertical angle of the given cone. Bisect the arc of the segment in F. Draw two lines FK, FK' to meet AA' produced both ways and such that, if they respectively meet the segment in Q, Q', FK : KQ = FK' : K'Q' = A A' : AL. DraAv QiV parallel to AA', and AN parallel to QF, meeting in N. Join AQ, A'Q, and let A'Q meet AN in R. Conceive a cone drawn with Q as apex and as bii.se the circle on AR as diameter and in a plane at right angles to that of AFA'. This cone will be such that the given ellipse is one of its sections. 220 THE COXICS OF APOLLONIUS. For, since FQ, AR arc parallel, ZFQR= ^ARQ, .•. zARQ^zFAA' = ζ OBG. And zAQR=zAFA' = ζ BOG. Therefore the triangles QAR, OBG are similar, and likewise the cones QAR, OBG. ΝοΛν A A' : AL = FK : KQ, by construction, = FK.KQ:KQ' = A'K.KA:KQ' = {A'K:KQ).(KA:KQ) = (QN : NR) .{QN: Ν A ), by parallels, = QN':AN.NR. Therefore [Prop. 3] AL is the latus rectum of the elliptic section of the cone QAR made by the plane of the given ellipse. And AL is the latus rectum of the given ellipse. Therefore that ellipse is itself the elliptic section. In like manner another similar right cone can be found with apex Q' such that the given ellipse is a section. No other right cone besides these two can be found satis- fying the given conditions and having its apex on the same side of the plane of the given ellipse. For, as in the preceding proposition, its apex P, if any, must lie on the arc A FA'. Draw PM parallel to A'A, and A' Μ parallel to FP, meeting in M. Join AP, A'P, and let A Ρ meet A' Μ in S. The triangle PA'S will then be similar to OBG, and we shall have PM' : A'M. MS= AT. Τ A' : TP^ = FT. TP : TP\ in the same way as before. We must therefore have AA' : AL = FT: TP ; and this is impossible, because AA'.AL = FK:KQ. VALUES OF CERTAIN FUNCTIONS OF THE LENGTHS OF CONJUGATE DIAMETERS. Proposition 123 (Lemma). [VIL 1.] In a parabola*, if PN be an ordinate and AH be vieusnred along the aocis a^uay from Ν and equal to the latus rectum, AP' = AN.NH. [=AN{AN + p„)] This is proved at once from the property PN^ = p„ . AN, by adding AN^ to each side. Proposition 124 (Lemma). [VII. 2, :l] If A A' be divided at H, internally for the hyperbola, and exte^'n ally for the ellipse, so that AH : HA' = p„•. AA\ then, if PN be any ordinate, AP':AN.NH=AA'.A'H. * Though Book VII. is mainly concerned with conjuKato diameters of a central conic, one or two propositions for the parabola are inserted, no doubt in order to show, in connection with particular propositions about a central conic, any obviously correspondinR properties of the parabola. 222 THE COXICS OF APOLLONIUS. Pro(iuce Λλ^ to K, so that ΛΝ.ΝΚ = Ρλ'^*; thus AN.NK.AN.A'N = PN':AN.A'N = p„: A A' = AH : A'H, by construction, or NK:A'N = AH.A'H. [Prop. 8] It folloAvs that A'N ±NK : A'N = A'H ± AH : A'H (where the upper sign applies to the hyperbola). Hence A' Κ : A'N=AA' : A'H; .•. A'K ±AA' : A'N ±A'H = AA' : A'H, or AK:NH = AA':A'H. Thus AN.AK:AN.NH = AA':A'H. But AN.AK=AP\ since AN.NK = PN\ Therefore AP^ : AN.NH = A A' : A'H. The same proposition is true \ΐ AA' is the minor a.xis of an ellipse and ;>„ the corresponding parameter. LENGTHS OF TONJUOATE DIAMETERS. 223 Proposition 125 (Lemma). [VII. 4.] If in a hyperhnUt or an ellipse the tangent at Ρ meet the aa-ift Λ A' in T, and if OD be the semi-diameter pfarallel to PT, then ΡΓ : CD' = NT : CN. Draw AE, TF at right angles to CA to meet GP, and let A Ε meet PT in 0. Then, if ρ be the parameter of the ortlinates to PP', we have ^.PT=OP:PE. [Prop. 23] Also, since CD is parallel to PT, it is conjugate to CP. Therefore ^.CP = CD' (1). Now OP :PE=TP:PF; .•. %.PT = PT.PF, .PF = Pr From (1) and (2) we have ΡΓ : CD^ = PF:GP = NT -.CN. .(2). 224 THE COXICS OF APOLLONIUS. Proposition 126 (Lemma). [VII. 5.] In a parabola, if ρ he the parameter of the ordinates to the diameter through P, and Ρ X the principal ordinate, and if AL he the latus rectum, p^AL + 4>AN. Let the tangent at A meet PT in and the diameter through Ρ in E, and let PG, at right angles to PT, meet the axis in G. Then, since the triangles PTG, EPO are similar, GT:TP=OP:PE, ' ' 2 l^^^f ^-j Again, since TPG is a right angle. TN.NG = PN^ = LA.AN, by the property of the parabola. But TN=2AN. [Prop. 12] Therefore AL = 2NG (2); t bus AL^ ^AN= 2 (TN + NG) = 2TG = p, from (1) above. LENGTHS OF CONJUOATE DIAMETERS. 22.') [Note. The property of the normal (iV(V = halt' th.• latus rectum) is incidentally proved here by regarding it as the perpendicnlar through Ρ to the tangent at that point. Cf. Prop. 85 where the normal is regarded as the mininnim straight line from G to the curve.] Def. If AA' be divided, internally for the hyperbola, and externally for the ellipse, in each of two points H, H' such that A'H : AH= AH' : A'H'=AA' : p^, where pa is the parameter of the ordinates to A A', then AH, A'H' (corresponding to pa in the proportion) are called homologues. In this definition A A' may be either the major or the minor axis of an ellipse. Proposition 127. [VII. 6, 7.] //" AH, A'H' he the " hoDwlogues" in a hypei'bola or an ellipse, and PP', DD' any two conjugate diameters, and if AQ he draivn parallel to DD' meeting the curve in Q, and QM he perpendicular to AA' , then PP" : DD" = MH' : MH. Join A'Q, and let the tangent at Ρ meet A A' in T. Then, since A'C= CA,2a\aQV= VA (where GP meets QA in V), A'Q is parallel to CV. Now ΡΓ ■.CD' = NT : CN [Prop. 1 2ό] = AM : A'3i, by similar triangles. And, also by similar triangles, CP':Pr = A'Q':AQ\ whence, ex aeqiiali, CP' : CD' = (AM : A'M) . (A'Q' : AQ') = (AM : A'M) X {A'Q' : A'M . MH') X (A'M.MH' : AM. MH) χ (.1^/ . MH : AQ'). H. c. I '• 226 THE C'OXICS OF Al'OLLONIUS. LENGTHS OF COX.TUOATE DTAMKTFRS. 227 But, by Prop. 124, Λψ•.Λ'Μ.ΜΗ' = ΑΑ':ΛΗ', and AM.MH :AQ' = A'H : AA' = AH' : A A'. Also A'M.MH' : AM. Μ Η =(A'M : AM) . (MM' -. Μ Η ). It follows that CP' : CD"" = MH' : MH, or PP" : DD" = MH' : MH. This result may of course be written in the form PP' : ρ = MH' : MH, where ρ is the parameter of the ordinates to PP'. Proposition 128. [VII. 8, 9, 10, 11.] In the figures of the last proposition the follovnng relations hold for both the hyperbola and the ellipse : (1) A A'•' : {PP' + DD'f = A' Η . MH' : {MH' ± '^MH.MHJ, (2) AA'•' : PP' . DD' = A'H: x^MH.MH', (3) AA'' : {PP" ± DD") = A'H : MH+ MH'. (1) We have AA" : PP" = CA' : CT' ; .•. AA" : PP" = CN. GT : GP' [Prop. 14] = A'M. A' A : A'q\ by similar triangles. Now A'Q' : A'M. MH' = A A' : AH' [Prop. 124] = AA':A'H = A'M. A A: A'M. A'H, whence, alternately, A'M. A' A : A'Q' = A'M. A'H : A'lM . MH'. Therefore, from above, AA":PP" = A'H:MH' (a), = A'H.MH': MH'\ 15—2 228 THE COXICS OK APOLI.OXirs. Again, PP" : DD" = MH' : MH ... {β), [Prop. 127] = MH"':MH.MH'• PP' '.DD' = MH' : \/MH . MH' (7). Hence PP' : PP' ± DD' = MH' : MH' + \'MH . MH', and PP" : (PP' ± DD'f = MH" : {MH' ± ^MH.MH'f . Therefore by (a) above, ex aeqmdi, A A" : {PP• ± DD'f = A'H.MH' : {MH' + ^MH.MH'f. (2) We derive from (7) above PP" : PP' . DD' = MH' : ^MWTMH'. Therefore by (a), ex aequal A A" : PP' . DD' = A'H : s/MH.MH'. (3) From {β), PP" : {PP " ± DD") = MH' : MH ± MH'. Therefore by (a), ex aequali, AA" : {PP" + DD") = A'H : MH + MH'. Proposition 129. [VII. 12, 13, 29, 30.] /?? every ellipse the sum, and in every hyperbola the difference, of the squares on any two conjugate diameters is equal to the sum or difference respectively of the squares on the axes. Using the figures and construction of the preceding two propositions, we have AA" : BB" = AA' : p« = A'H : AH, by construction, = A'H -.A'H'. Therefore A A" : A A" ± BB" = A'H : A'H ± A'H' (where the upper sign belongs to the ellipse), or AA".AA" + BB" = A'H:HH' (a). LENGTHS OF COXJLTgaTK DIAMKTEHS. 220 Again, by (a) in Prop. 128 (1), AA'':FF" = A'H:i]IH', and, by means of (β) in the same proposition, FF" : {FF" + DD") = MH' : MH ± MH' = MH'.HH'. From the hist two relations we obtain AA" : {FF"±DD") = A'H : HH'. Comparing this with (a) above, we have at once Proposition 130. [VII. 14, 15, 16, 17, 18, 19, 20.] Tlie following results can be denved from the preceding proposition, viz. (1) For the ellipse, A A" : FF" ~ DD" = A'H : 2CJ/; and for both the ellipse and hyperbola, if ρ denote the parameter of the ordinates to FF', (2) AA" : p' = A'H. MH' : MH\ (3) AA" : {FF'±pY = A'H . MH' : {MH ± MH'f, (4) AA" :FF'.p = A'H : MH, and (5) AA":FF" + p' = A'H.MH' : MH" ± MH\ (1) We have AA"' : FF'^ = A'H : MH', [Prop. 128 (1), (a)] and FF" : FF" - DD" = MH' : MH' ~ Μ Η [ibid., {β}] = ΜΗ' : 2CM lu the ellipse. Therefore for the ellipse AA":FF"-- JJJJ" = A'H : 2CM/. 230 THE coyics of apollonius. (2) For either curve ΑΛ" : PP' = A'H : MH', as before, = A'H.MH':MH'\ and, by Prop. 127, PP'':f = MH"':MH'•, .•. AA" : p' = A'H.MH' : MH\ (3) By Prop. 127, PP' -.ρ^ΜΗ'.ΜΗ•, .•. PP" : {PP' ±ργ = ΜΗ" : (ΜΗ ± MH')\ And ΑΑ":ΡΡ'•' = ΑΉ. ΜΗ' : MH'\ as before ; .•. AA" : (PP' ±pf = A'H . MH' : (MH + MH'y. (4) A A" : PP' = A'H : MH', as before, and PP".PP'.p = PP' : ρ = MH'.MH; [Prop. 127] .•. AA'':PP'.p = A'H:MH. (5) AA" :PP" = A'H. MH' : MH", as before, and PP" : PP" ± p' = MH" : MH" ± MH\ by means of Prop. 127 : :. AA": PP" ± if = A'H. MH' : MH" + MH\ Proposition 131. [VII. 21, 22, 23.] In a hyperbola, if AA' ^^.J BB', then, if PP', DD' he anij other two conjufjute diameters, Ρ P' ^^^ DD' respectively ; and the ratio PP' : DD' continually \ . ^ > as Ρ moves "^ (or increases J farther from A on either side. Also, if AA' = BB', PP' = DD'. LENGTHS OF CONJUGATE DIAMETERS. 2^1 (1) Of the figures of Prop. 127, the first corresponds to the case where AA' > BB', and the second to the case where AA'kBB'. Taking then the \ ^ Λ figure respectively, it follows from PF'' .DD'* = MH' : MH [Pn.p. 1 27] that PP' ^j.> DD'. Also AA '' : BB"' = A A' : pa = A'H : AH, by construction, = AH' : AH, and AH' : AH ^^> MH' : MH, while MH' : MH \ . \ continually as Μ moves further (or increases] "^ from A, i.e. as Q, or P, moves further from A along the curve. Therefore AA" : BB'\^.^ PP" : DD'\ and the latter ratio \ ^ i as Ρ moves further from ^4. (or mcreasesj And the same is true of the ratios AA' : BB' and PP' : DD'. (2) ΙΪ AA' = BB', then AA'=pa, and both Η and //' coincide with G. In this case therefore AH = AH' = AG, MH = MH' = GM, and PP' = DD' always. Proposition 132. [VII. 2+.] In an ellipse, if A A' be the imijor, and BB' the minor, a-ris, and if PP', DD' be any other two conjugate diameters, then AA' : BB' > PP' : DD', and the latter ratio diminishes continually as Ρ moves from A to B. 232 THE COXIC.S OF APOLLUNIUS. We have CA' : CB' = AN . λ'Α' : PN' ; .•. AN.NA'>PN\ and, adding C'iV"^ to each, CA' > CP\ or AA'>PP' (1). Also GB' : CA' = BM. MB' : DM' where DM is the ordinate to BB'. Therefore BM . MB' < DM\ and, adding CM\ GB' < GD' ; .•. BB'kDD' (^)• Again, if P^P^, D^D^ be another pair of conjugates, P, buing further from A than P, D, will be further from Β than D. And AN. Ν A' : AN^ . N,A' = PN' : P^N;". But AN^.N^A'>AN.NA'; .•. p,n;'>pn\ and AN^ . N^A' - AN . Ν A' > P^N^' - PN\ But, as above, AN^ . N^A ' > P,N^\ and AN^ . N^A '-AN. Ν A ' = GN' - CiY," ; .•. CN' - GN^' > P^N;" - PN' ; thus GP•' > GP^\ or PP'>PJ\' (3). In an exactly similar manner we prove that DD' <D^D; (4). LENGTHS OF CONJUGATK niAMETKKS. 238 We have therefore, by (1) and (2), AA'.BB'>PP'.DB', and, by (3) and (4), FP' : DD' > PJ\' : D^D^'. Cor. It is at once clear, if pa, p, /), are the parameter corresponding to A A', PP', PyP^y that Pct<p, P<p„ etc. Proposition 133. [VII. 25, 26.] (1) In a hi/perbola or an ellipse AA' + BB'<PP' + DD', where PP\ DD' are any conjugate diameters other than the axes. (2) In the hyperbola PP' + DD' increases continually as Ρ moves further from A, while in the ellipse it increases as Ρ moves from A until PP', DD' take the position of the equal conjugate diameters, lulien it is a maximum. (1) For the hyperbola AA" ~ BB" = PP" ~ DD" [Prop. 129] or {AA' -\- BB') . (^.4' ~ BB') = ( PP' + DD') . (PP' ~ DD '), and, by the aid of Prop. 131, AA' ~ BB' > PP' ~ DD' ] .•. AA' + BB'<PP' + DD'. Similarly it is proved that PP' + DD' increases as Ρ moves further from A. In the case where AA'^BB', PP' = DD', and PP'>AA• . and the proposition still holds. (2) For the ellipse AA' : BB' > PP' : DD' ; .•. {A A" + BB") : {AA' + BB')' > {PP" + DD") : {PP' + DD'f.* But AA"+BB" = PP" + DD": [Prop. 12!>] .•. AA' + BB'<PP' + DD'. * ApoUouius draws this inference directly, iind gives no intenuediute stt'pe. 234 THE COXICS OF Al'OLLONIUS. Similarly it may be proved that PP' + DD' increases as Ρ moves from Λ until PP', DD' take the position of the equal conjugate diameters, when it begins to diminish again. Proposition 134. [VII. -27.] J II every ellipse or hyperbola having unequal axes AA''-BB'>PP' -DD', luliei'e PP', DD' are any other conjugate diameters. Also, as Ρ moves from A, PP' - DD' diminishes, in the hyperbola con- tinually, and in the ellipse until PP', DD' take up the position of the equal conjugate diameters. For the ellipse the proposition is clear from Avhat was proved in Prop. 132. For the hyperbola AA" - BB" = PP" ~ DD", and PP'>AA'. It follows that AA' ~BB'>PP' ~DD', and the latter diminishes continually as Ρ moves further from A. [This proposition should more properly have come before Prop, 133, because it is really used (so far as regards the hyperbola) in the proof of that proposition.] Proposition 135. [VII. 28.] In every hyperbola or ellipse AA' . BB' < PP' . DD', and PP' .DD' increases as Ρ moves aiuay from A, in the hyperbola continually, and in the ellipse until PP', DD' coincide with the equal conjugate diameters. Wc have AA' + BB' < PP' + DD', [Prop. 133] so that .•. (A A' +BB'y < (PP' + DD'f. LENGTHS OF CONJUOATK 1)ΙΛΜ1•:ΤΚΙ{> •235 And, for the ellipse. AA"+ BB" = PP'' + Dl)"'. [Prop. 1 2!)] Therefore, by subtraction, AA' .BB' <PP' .DD\ and in like manner it will be shown that PP . DD' increases until PP', DD' coincide with the equal conjugate diameters. For the Ityperhola [proof omitted in Apollonius] PP' > A A', DD' > BB', and PP', DD' both increase continually as Ρ moves away from A. Hence the proposition is obvious. Proposition 136. [Vll. :u.] If PP', DD' be two conjugate diameters in an ellipse or in conjugate hyperbolas, and if tangents be drawn at the four extremities forming a parallelogram LL'MM', then the parallelogram LL'MM' = red. A A' . BB'. Let the tangents at P, D meet the axis AA' in T, T' respectively. Let Ρ Ν be an ordinate to A A', and take a length PO such that PO' = aN.NT. Now CA' : GB' = CN . NT : PN' [Prop. 1 4] = PO''.PN\ or ΟΑ.ΟΒ = Ρ0:ΡιΎ: .•. CA' : CA . CB = PO . C'P : CT . PN. Hence, alternately, CA' : PO.CT = CA . GB : GT . PN, or CT.CN:PO.CT=GA.GB.GT.PN (1). 236 THE CONICS OF Al'OLLONIUS. Again, ΡΓ : CD' = NT : CN, [Prop. 125] so that 2 Δ CPT : 2 Δ Τ' DC = NT : CN. But the parallelogram (CL) is a mean proportional between 2 Δ CTT and 2 A Τ DC, for 2ACPT:(CL) = PT:CD = CP : DT' = {GL)'.2AT'DC. Also PO is a mean proportional between CN and iVT. Therefore 2ACPT : (CZ) = PO : CN = PO . CT : CT . CN = CT.PN : CA . CB, from (1) above. And 2ACPT=CT.PN; .•. (CL) = CA . CB, or, quadrupling each side, CJLL'MM'^AA'.BB'. Proposition 137. [VII. 3:}, S^, :3.5.] Supposing pa to be the parameter corresponding to the axis A A' in a Jujperhola, and ρ to be the parameter corresponding to a diameter PP', (1) if A A' he not less than p^, then p„ < p, and ρ increases continually as Ρ moves farther from A ; LENGTHS OF CONJUOATR DIAMETERS. 2ii7 (2) if A A' he lesa than p„ but not less than '-^' , then p,, < p, and ρ increases as Ρ moves away from A ; (0) if AA' < -^ , there can be found a diametei' Ρ,^Ρή on either side of the aa-is suck that p^='2P^P^. Also p» is less than any other parameter ρ , and ρ increases as Ρ moves further from Po ill either direction. (1) (o) ΙϊΑΑ'=ρα, we have [Prop. 131 (2)] PP'=p = I)D\ and PP', and therefore p, increases continually as Ρ moves away from A. (b) If AA'>pa, AA'>BB', and, as in Prop. 131 (1), PP' : DD', and therefore PP' : p, diminishes continually as Ρ moves away from A. But PP' increases. Therefore ρ in- creases all the more. (2) Suppose AA'<pa but ^^. Let Ρ be any point on the branch with vertex A ; draw A'Q parallel to CP meeting the same branch in Q, and draw the ordinate QM. Divide A' A at H, H' so that A'H : HA = AH' : H'A' = A A' : pa, as in the preceding propositions. 238 THE COXIC'S OF AI^OLLONIUS. Thcreiuic ΛΛ" : pa' = Λ'Η . ΛΙΓ : AH' (a). We have now AH > AH' but iif 2AH'. And MH+HA>2AH; .•. MH+HA. AH>AH:AH', or iMH+HA)AH'>AH' (β). It follows that {MH + HA) AM : (MH + HA) AH', or AM : ^F', Therefore, componendo, MH' .AH'< (MH + HA) AM+AH' : AH' <MH':AH' (7), whence A'H.MH' : A'H.AH' < MH' : AH\ or, alternately, A' Η . MH' : MH' < A'H.AH': AH\ But, by Prop. 130 (2), and by the result (a) above, these ratios are respectively equal to AA" : p', and A A" : pa. Therefore AA" : p' < AA" : pa\ or Pa<P- Again, if Pj be a point further from A than Ρ is, and if A'Q^ is parallel to CP^, and il/, is the foot of the ordinate Qil/,, then, since AH :|* 2AH', MH < 2MH' ; also M^H + HM>2MH. Thus (ιΜ,Η + HM) MH' > MH\ This is a similar relation to that in (/3) above except that Μ is substituted for A, and M^ for M. We thus derive, by the same proof, the corresponding result to (γ) above, or M^H'.MH' <M^H'.MH\ whence A' Η . M^H' : M^ H' < A' Η . MH' : MH\ or AA'^ : p^' < A A" : p\ so that /) < p^ , and the proposition is proved. LENGTHS OF ΓΟΝΜΓΓίΑΤΕ DIAMKTKllS. •2ίί9 (3) Now let ^^' be less than 2 • Take a point .1/,, such that HH' = Η'Μ^, and let Q,., 1\ be related to Mo in the same way that Q, Ρ are to il/. Then PoPo' : Po = M,H' : M,H. [Pn.,,. 127] It follows, since HH' = H'M„, that P,= 2P,P:. Next, let Ρ be a point on the curve between P„ and ^l, and Q, Μ corresponding points. Then M,H'.H'M<HH'\ since MH'<M,H'. Add to each side the rectangle {MH + HH) MH', and we have (MM+HAI)MH'<iMH\ This again corresponds to the relation (β) above, with Μ substituted for Λ, M^ for M, and < instead of >. The result corresponding to (7) above is MoH':MH'>MoH'.MH': .•. ΑΉ.Μ,Η' : M,W > A'H. MH' : MH\ or AA'':p:>AA":2f. Therefore ρ >pn• And in like manner we prove that jj increases as 7^ moves from Po to ^. Lastly, let Ρ be more remote from A than P^ is. In this case H'M > Η 'Mo, and we have MH' . H'M^ > HH", and, by the last preceding proof, interchanging Μ and Mo and substituting the opposite sign of relation, AA" : p' < AA" : po\ and p>Po• In the same way we prove that ρ increases ai> Ρ moves further away from Ρ and A. Hence the proposition is established. •240 THE CON/rS OF APOLLONIUR, Proposition 138. [VII. 36.] In a hyperbola witli unequal axes, if pa he the parameter corresponding to A A' and ρ that corresponding to PP', AA' -pa>PP''P, and PP' - ρ diminishes continually as Ρ moves away from A. With the same notation as in the preceding propositions, A'H : HA = AH' : H'A' = AA' : p„, whence A A" : (A A' ~ paf = A'H. AH' : HH". Also [Prop. 130 (3)] A A" : (PP' ~ pY = A'H. MH' : HH'\ But A'H.MH'>A'H.AH'; .•. AA'- : (PP' - pY >AA": (A A' - p„f. Hence AA' ~ ])„> PP' - p. Similarly, if P,, M^ be further from A than P, Μ are, we have A'H.M^H'>A'H.MH', and it follows that PP''-p>P^P^' ^p,, and so on. Proposition 139. [VII. 37.] In an ellipse, if P^Po, Df^D,'hethe equal conjugate diameters and PP', DD' any other conjiigate diameters, atid if po, p, Pa, Pb he the parameters corresponding to PqPO, PP', A A', BB' respectively, then (1) AA' ~ Pa is the maximum value of PP' - ρ for all points Ρ hetween A and P^, and PP' - ρ diminishes con- tinually as Ρ moves from A to Po, LENGTHS OF CONJUGATE DIAMETERS. 241 (2) BB' - pi, is the maximum value of PP' - ρ for all points Ρ between Β and 2\,, ami ΓΓ' - ρ diminishes continually as Ρ passes from Β to Po, (3) BB'-pu>AA'-pa. The results (1) and (2) follow at once from Prop. 182. (3) Since pb : BB' = A A' : y)„, and pt, > A A', it foiiow.s at once that BB' -^ pi,> AA' ~ pa. Proposition 140. [VII. 38, 39, 40.] (1) In a hyperbola, if A A' be not less than I j)„, PP' + p >AA'+pa, luhere PP' is any other diameter and ρ the corresponding parameter; and PP'+j) will he the smaller the nearer Ρ approaches to A. (2) If AA' <^p)a, there is on each side of the axis a diameter, as PqPo, such that P^Po' = ^Po ; (f>i(l Ρ,^Ρ^'+Ρο is less than PP' + p, where PP' is any other diameter on the same side of the axis. Also PP' + p increases as Ρ moves away from P.. (1) The construction being the same as before, we suppose (ft) AA'-^pa. In this case [Prop. 137 (1)] PP' increases as Ρ moves from A, and ρ along with it. Therefore PP' + ρ also increases continually. (b) Suppose AA' <pah\\t -i^^pa', .•. AH'^\AH\ thus AH'-^liAH + AH'), and {AH + ΑΗ')ΛΑΗ' -^{AH ^ AH')\ Hence 4>{AH+AH')AM ■Α{ΑΗ+ΑΙΓ)ΑΗ', or AM:AJI', ^^{AH + AID AM : (.1 // + Λ ΙΙ'Ϋ ; Η. C, lt> 242 THE COXICS OF APOLLONIUS. and, componendo, ΜΗ':ΑΗ'^^ΑΗ + ΑΗ')ΛΜ+(ΛΗ + ΛΗΎ•.(ΛΗ + ΑΗ'γ. Now (3iH + MH'f -{AH + AHy = 2AM(3IH + MH' +AH +AH') >4>AM{AH + AH'); .•. 4A3T(AH + AH') + (AH + AHy<{]\IH + MHy. It follows that MH' : AH' < (MH + MH'y : (AH + ΑΠγ, or A'H.MH' : {MH + MHJ < A'H .AH': {AH + AH J ; .•. AA'':{PP' + pf<AA":{AA' + py [by Prop. 180(3)]. Hence AA' + pa< PP' + 2^• Again, since AH'i 1{AH + AH'), MH'>l{MH + MH'); .• . 4 {MH + MH') MH' > {MH + MH')\ And, if Pj be another point further from A than Ρ is, and Qi , il/, points corresponding to Q, M, we have, by the same proof as before (substituting Μ for A , and il/j for M), Α'Η.Μβ' : {M^H + M^HJ < A'H.MH' : {MH + MHJ. We derive PP'+p< P,P^+p, ; and the proposition is established. (2) We have AH' < ^AH, so that AH'< \HH'. Make H'M^ equal to ^HH', so that MoH' = ^MoH. Then P,P: : ^Jo = M^H' .M,H=l:S, and PoPo' = f. Next, since -Λ/ο-ί'^' = i -^^o-H", M,H'=i{M,H + M,H'). Now suppose Ρ to be a point between A and Po, so that il/„7/'>il/ii"; .•. {MoH+M,H'f > {M^H + MH') . 4il/„^'. LENGTHS OF COXJUOATE DIAMETERS. 243 Subtracting from each side the rectangle (M^H + }ΙΗ')ΛΜΜ„, (ΜΗ + MH'y > (MoH + MH') . ^MH' ; .•. {M,H + MH') . 4il/il/„ : {M,H ^ MH') . 4MH', or MM.. : Mil', >{MJI+MH')AMM.. : (ΜΗ + ΜΠ')\ Therefore, componendo, Μ,Η': MH'>(M,H+MH') . 4MM,+{MH+MH'y -. {MH+MH'f > (MoH + MoH'f : (MH + MH'f. Hence A'H.MoH' : {M,H+M,Hy > ΛΉ . MH' : {MH + MH'f. Tlierefore [Prop. 130 (3)] AA" : (ΡοΛ' + i>o)' > AA" : {PP'-^pf, and PP' + p>1\P:+p,. Again, if Pi be a point betΛveen Ρ and A, we have (MH + MH'f > (MH + ilA H') . ^MH', and we prove exactly as before that P,P;+p,>PP' + p, and so on. Lastly, if Μ Η > M„H, we shall have (MH + MM') . ^M, H' > ( J/„ // + Μ,Η'Υ. If to both sides of this inequality there bo added the rectangle (MH + Μ,,Η') ■ ^fMM^, they become respectively (MH + M,H') . ^MH' and (.1/// + MH')\ and the method of proof used above gives PoPo' + p«<PP'+/>, and so on. Hence the proposition is established. IG— 2 244 THR COXICS OF APOLLONIUS. Proposition 141. [VII. 41.] In any ellipse, if PP' be any diameter and ρ its parameter, PP' -\-p> AA' -{•ρ>α, and PP' + ]) is the less the nearer Ρ is to A. Also ΒΒ'Λ-ρι>ΡΡ' + ρ. Q, With the same construction as before, A'H.HA = AH''.H'A' = AA''.p, = p,:BB'. Then A A" : {AA' + 2^αΥ = Λ'Η-' : HH" = A'H.AH' '.HH'"- (a). Also AA'^:BB" = AA':pa = A'H:A'H' \ = A'H.A'H':A'H" i. and BB" : {BB' + p^f = A'H" : HH" J Therefore, ex aequali, AA"'.(BB' + p(,f = A'H.A'H':HH'' (β). From (a) and (β), since AH' > A'H', AA' + pa<BB' + pi,. Again AA":{PP' + pf = A'H.MH' : HH'\ [Prop. 130 (3)] and A A" : (ΛΡ/ + ρ,γ = A'H . M,H' : HH'\ Λvhere Pj is between Ρ and B, from which it follows, since AH' > MH' > M,H' > A'H', that AA'-\-2)a<PP'+P, ΡΡ' + ρκΡ,Ρ,' + ρ,, P,P,' + p,<BB' + p,, and the proposition follows. LENGTHS OF CONMrcATI•: mAMKTKKS. 24.") Proposition 142. [VII. 42.] //; a hyperbola, if PP' be any diameter luith parameter p, AA'.pa<PP'.p, and PP' .p increases as Ρ moves away from A. We have A'H : HA = A A" : AA'.pa, and A'll : Μ Η = A A'' : PP'.p, [Prop. 1 30 { 4 )] while AH<MH• .•. AA'.pa<PP'.p, ;aid, since MH increases as Ρ moves from A, so does PP'.p. Proposition 143. [VII. 43.] In an ellipse AA'.pa< PP'.p, where PP' is any diameter, and PP'.p increases as Ρ moves aivay from A, reaching a maximum luhen Ρ coincides with Β or B'. The result is derived at once, like the last proposition, from Prop. 130 (4). [Both propositions are also at once obvious since PP'.p = DD'\] Proposition 144. [VII. 44, 45, 4ϋ.] In a hyperbola, (1) if A A' ^ Pa, or (2) if AA' < Pa, but AA"^}, {A A' - p„)\ then AA" + Pa'<PP" + p\ where PP' is any diameter, and PP'^ + j)^ increases as Ρ moves away from A ; 24U THE COXICS OF APOLLONIUS. (3) if ΛΛ'^ < ^(ΛΑ' ~ paT, then there will he found on either side of the a. ris a diameter PoP» such that PqPo^ = hi^oPo " PoT, and Ρ,Ρο" + ρό" will he less than PP'^ + p\ where PP' is any other diameter. Also PP''- + p^ will he the smaller the nearer PP' is to PoPJ. (1) Let AA' be not less than pa- Then, if PP' be any other diameter, ρ > pa, and ρ increases as Ρ moves further from A [Prop. 137 (1)]; also AA' <PP', which increases as Ρ moves further from A ; .•. AA"-^pa'<PP"+p\ and PP'^ + p'^ increases continually as Ρ moves further from A. (2) Let A A' be less than pa, but A A" ^ | (A A' ~ paf. Then, since AA' : j^a = ΛΉ : AH = AH' : A'H', 2AH"^HH'\ and niH'.AH' >HH'\ Adding 2AH .AH' to each side of the last inequality, 2{MH + AH')AH'>2AH.AH'+HH" >AH' + AH"•, .•. 2{MH+AH')AM:2{MH + AH')AH', or AM . AH', < 2 (MH + AH') AM : AH' + AH'\ Therefore, componendo, MH':AH'<2{MH + AH')AM + AH' + AH".AH' + AH'\ and MH' + MH" = AH' + AH" + 2AM {MH + AH'), so that MH' : AH' < MH' + MH" : AH' + AH", or ΛΉ .MH' : MH' + MH" < A'H .AH' : AH' + AH"; .•. AA":PP" + p'<AA":AA" + pa'. [Prop. 130(5)] Thus AA"-\-pa'<PP"+p'. Again, since 2MH" > HH", and (if AM, > AM) 2M,H'. MH' > HH", we prove in a similar manner, by substituting Μ for A and il/, for M, that LEN(;THS OF COXJUOATE DlAMKTKIiS. 247 (3) Let ΛΛ' bo less than ^{AA' - /)„)', so that 2.Air<III['\ Make 2MoH'' wiual to HH'\ Now M,H' : M,H = Pol\' : p„ [Γη .p. 1 27] so that Ρ,Ρο" = i (PoPu' ~ }\y. Next, if Ρ be between A and Po> 2.1/oif "* = HH", and 2M,H'.MH'<HH'\ Adding 2MH.MH' to each side, 2 {M,H + MH') MH' < MIP + MH'\ and, exactly in the same way as before, we prove that ΡοΡο'•-•+Κ<ΡΡ"-' + /. Again, if Pj be between A and P, whence (adding 2M,H.M,H') 2 (iViT + M,H') M,H' < M,H' + Μ,Η', and, in the same Avay, ΡΡ'•^+/<ΛΡ/^+Κ• Similarly ΛΡ/'^ + ρ,' < ^.1'^ + pa\ Lastly, if AM > AM^, 2MH'.iMoH'>HH", and, if AM, > AM, 2M,H'.MH'>HH"; whence we derive in like manner that PP''+/>PuP..'-' + iV, PJ\"+p;'>PP"+p\ and so on. 248 THE cosies OF AlOLLOXIUS. Proposition 145. [VII. 47, 48.] In an (1) if A A" -if \{AA' + pa)\ then ΑΛ" + pa' < PP" + p\ and the latter increases as Ρ moves away from A, reaching a maximum when Ρ coincides with Β ; (2) if AA'^ > ^{AA' + paf, then there luill be oii each side of the accis a diameter PqPo such that PqPo'^ = ^{PoPo + 2>ν)\ and ΡαΡά^ λ- pn will then he less than ΡΡ''^Λ -p^ in the same quadrant, while this latter increases as Ρ moves fy^om Pq on either side. (1) Suppose AA":i(^^{AA'+2)af• ΝοΛν A'H. AH' : ΑΉ' + A'H" = AA" : AA" +Pa'• Also AA" : BB" = pi, : BB' = A A' : p^ = A'H : A'H' = A'H. A'H' : A'H", and BB" : (BB" +pi,') = A'H" : A'H' + A'H'' ; hence, ex aequali, A A" : {BB" + pi') = A'H . A'H' : A'H' + A'H'\ and, as above, AA" : {AA'-'+pa') = A'H. AH' : A'H' + A'H". Again, AA":i^^{AA'+pJ, .•. 2A'H.AH'^HH'\ whence 2 A' Η . ΜΗ' < Η Η". Subtracting 2ΜΗ . ΜΗ', we have 2A'M.MH'<MH' + MH" (1), .•. 2A'M.AM : 2A'M. MH', or AM : MH', >2A'M.AM -.MH' + MH", and, since 2.1'.!/ . AM + MH' + MH" = A'H' + A'H", LENGTHS OF CONJUfJATK DIAMETKUS. 24!) we have, compunendo, AH' : MH' > A'H' + A'W' : MIP + MH'\ .•. A'H.AH' : A'H' + A'H" > A'U.MW : MIP + MH'\ whence A A"' : (^1^1'" + ^v) > A A '= : {PP" -f /), [Ριυρ. 130 (.->)] Again, either Μ Η < M,H\ ur J/i/.^ M,H'. (a) Let MH<M,H'. Then J/i/-^ + J/7/'^ > J/.y/^ + M,H'\ and J/jiT' + MJi" > M,H' ■ 2 {MJi' - iV//)* ; .•. JAUi • ■2{M,H'- MH) : JAii'. 2 (J/^/i '- J///), or MM, : .1/. //', > MM, . 2 (i/iZT' - MH) : i/,^^ + J/, H'\ But il/if^ + il/^'* - (M, H' + M,H") = 2 {CM* - CM;') ; .•. il/J/i . 2(Λ/ιί^' - MH) + M,H" + M,H" = J//P + il///" ; thus, componendOy we have MH' : M,H'>MH' + MH" : M,H' + M,H"• therefore, alternately, A'H .MH' : MH' + MH" > A'H .M,H' : MJP + MJP\ and yl^'^ : PP"^ +/ > ^1^'^ : ΛΛ'^ +^ίΛ [Prop. 130 (ό)] so that ΡΡ" + ρ'<Ρ,Ρ,"+ρ'• (b) If MH<^M,H\ MH' + MH" ^ M, H' + M, H'\ and it results, in the same way as before, that A'H.MH' : MH' + MH">A'H.MJP : MJP + Mjr, and PP''+p-<PJ\''+p;'. Lastly, since ΑΉ.ΑΉ' : AH' + A'H" = AA" : BB'+po', and ΑΉ . MJP : M, H' + .1/.//'^ = A A" : i^ /^" + ih\ * As in (1) ftbove, .V, H- + .V, //'- > 2.1 '.U, . .1/,// ' > M^ir . 2 (.!/,//' - J/i/), u fortiori. 2')0 THE COMCS (JF Al'OLLONIUS. it is shown in the same nianiier that (2) Suppose AA'^ > h {A A' +p„)\ so that 2AH">HH'\ Make 2M,H" equal to HH", so that MM" = yiH" = HH' . CH' ; .•. Hir:Mjr = M,H' -OH' = HH' - M,H' ■■ Mjr ~ CH', whence M,H : CM, = //Zf ' : M,H', and //^' . (7J/o = M,H . M,H'• If then (a) AM < AM,, ^GMo.CH'>2MH.M,H'. Adding 2MMq.M,H' to each side, 4Cifo . CH' + 2il/il/o • il/oiT' > 2M,H . M,H', and again, adding '^CM^, 2 (C/il/ + CM,) M,H' > (Μ,Η' + Μ,Η"). It follows that 2 (CM + CM,) MM, : 2(CM + ClM,) M,H', or MM, : M,H', < 2 (6'i¥+ CM,) MM, : (ϋ/„ίΓ* + Μ,Η"). Now 2 (6'ϋ/ + CM,) MM, + il/o H' + .¥o^'' so that, componendo, MH' : il/o//' < MH' + i/i/'^ : M,H' + il/„^", and A'H.MW : MH' + MH"<A'H.M,H' : Μ,Η' + Μ,ΙΓ\ whence Ρ,Ρ," -\- p^' < PI'"' + p\ Similarly, if ^il/j < AM, •1HH'.CM>2M,H.MH', and we prove, in the same manner as above, pp''-^p^<pj>:-^^p;\ F,KN(!THS OF ΟΟΝΜΓίίΛΤΚ DIAMK IKKS. "Jol And. since 2////' . ( M/, > ΊΛ II . J/. // ', in like mauner Lastly (6), H AM > AM^, the same method of proof gives etc. Proposition 146. [Vll. 41), 50.] Ill a Ityperhola, (1) if A A' >pa, then A A" - Pa' < PP" - /, where PP' L• amj diameter, and PP" - / i)icreases as Ρ moves farther from A ; also PP" ~ p' > AA" ~ pa . AA' but < 2 (^ A'^' ~ p^ . AA') : (2) if A A' < Pa, then AA''^''Pa>PP''-^p\ which diminishes as Ρ moves away β'ΟΏΐ A ; also PP" ~ p' > 2 (^1^'^ « pa . AA'). (1) As usual, A' Η : ^i7 = AH' : ^'//' = xl^' : pa\ .•. A'H.AH' : ^if'^ - ^//^ = ^^" : .1.1'* ~ Pa\ Now iVif ' : ^ii' <MH.AH; .•. il///' : ^^' < MH' + Μ Η : AH' + AH < {MH' + MH) HH' : (.1//' + AH ) HH', i.e. < MH" - MH' : AH" ~ AH\ Hence A'H.MH' : ^1/^'* - il/^"^< A'H.AH' : .1//'^' ~ ^1//* ; .•. AA" : PP'" - p*< A A" : yl^"" ~ /;„'- [l*iop• 130 (5)] or ^^'•^-_p«''<PP"^-yr. Again, if AM^>AM, MJP.MH'kMJI -Mil- .•. MJP:MH'<MJl+ MJI : Mir + Mil, 252 THE aoxic.s of apollonius, ;ind, i)roccecling as befuro, wo find and so on. Now, if FO be measured along PP' eciual to }), PP"^p'=2P0.0P' + 0P"; .•. PP'' ~ / > PP' . OP' but < 2PP' . OP'. But PP'.uP' = PP"-PP'.PO = PP"-p.PP' = AA"-2)a . A A' ■ [Prop. 12!)] .•. PP" ~ ^/ > ^1^'^ ~ Pa . ^^' but < 2 (yl^'^ ~ p« . A A'). (2) If^lJ.'<jj„, il/i/':^/i'>il/i/:yliJ; .•. J\IB' : ^i/" > MH' + MH : ^iT' + AH, and ^'i/ . MW : ^'i/ . ^ii' > {MH' + MH)HH' : (^^' + ^ii) iiii', i.e. > MH" ~ il/Zf^ : ^^'^ ~ AH\ Therefore, proceeding as above, we find in this case PP"~p'<AA"'-pa\ Similarly and so on. Lastly, if PP' be produced to so that PO = p, AA"-pa.AA' = PP"'-p.PP' [Prop. 129] = PP'.OP'. And PP" -- if = PP" - PO' = 2ΡΡ'.ΡΌ + ΡΌ' >2PP'.0P' or > 2 (4.4'^ -^„. ^1^1'). .ENOTIIS OF COX.irOATE lHAMKTF.ltS. 2.')^ Proposition 147. [VII. 51.] In an ellipse, (1) if PP' he any diameter such that PP' > p, AA"--p„'>PP"^jf, and PP'^ - p^ diminishes as Ρ moves further from A ; (2) if PP' he any diameter such that PP' < p, BB" -^ Pf,' > PP'•' ^ p\ and PP'^ - p^ diminishes as Ρ moves further from B. {!) In this case (using the figure of Prop. 141) AH' : MH' < AC : CM .•. A'H.AH'.A'H.MH'< 2HH' .AC : 2HH' .CM i.e. < AH" ~ AH' : MH" ~ MH\ Therefore, alternately, A'H.AH' : AH" ~ AH' < A'H.MH' : MH' - MH\ Hence A A" : AA" ~ 2^a < AA" : PP" - p\ [Prop. 130 (5)] and AA"--pa'>PP"'-2}\ Also, if ^lil/j >AM, we shall have in the s;\jik• way A'H.MH': A'H.Mjr<MH''~ Mil-. MJI '- MJl\ and therefore PP" ~ / > PJ'" - p^, and so on. (2) Ρ must in this case lie between Β and the extremity of either of the equal conjugate diameters, and Μ will lie between C and A' if Ρ is on the (juadrant AB. 254 THE COXIVS OF APOLLONIUS. Then, if M^ corresponds to another point P,, and AAI^ > AM, we have MH'>M^H', and CM < CM^; .•. A' Η . ΜΗ' : ΑΉ . ili, Η' > CM : CM^ >2CM.HH':2CM^.HH', i.e. > MH' - MH" : M^H' ~ M^H'\ whence, in the same manner, we prove and PP'* - p^ increases as Ρ moves nearer to B, being a maximum when Ρ coincides with B. camiiripor: phintkd «y j. ανπ c. f. clay, at the university press. SOME PUBLICATIONS OF Diophantos of Alexandria; a Study in the History of Greek Algebra. By T. L. Hk.vth, M.A., Fellow of Trinity College, Cambridge. Demy 8vo. 7s. (id. Greek Mathematics, Short History of, by J. Gow, Litt.D., fonnerly Fellow of Trinity College. Demy 8vo. 10,*. G(/. Geometrical Conies. By F. S. Macaulay, M.A., A.ssistant Master at St Paul's School. Crown 8vo. 4s. ed. A Treatise on Abel's Theorem. By H. F. Baker, M.A., Fellow of St John's College, Cambridge, University Lecturer in Mathematics. Royal 8vo. [In the Prcjis. A Treatise on the Theory of Functions of a Complex Vaiiable. By A. R. Fousvrn, Sc.D., F.1\.S., Sadlerian l*rnfe.-^>..r .>f Pure Mathematics in the University of Cambridge anil Fellow of Trinity College. Royal 8vo. 21s. Net. A Treatise on Plane Trigonometry. By E. W. HonsoN, ScD., F.R.S., Fellow and Tutor of Christ's College. Demy 8vo. 12s. A Treatise on the Lunar Theory. By E. W. Buown, >Γ.Α., Fellow of Christ's College, Caml)ridge, I'rofessor of Applieil Mathe- matics in Haverford College, Penn.sylvania. Royal 8vo. 15s. Mathematical Papers ot the hite Arthur Cayley. Sc.D., F.R.S., Sadlerian Profes.sor of Pure Mathematics in the Univerxity of Cambridge. Demy 4to. To l)c completcil in 13 vols. Volumes I. TI. ITT. TV. Y. VT. VII. VIIl. and IX. 2.')*. each. [\Ό1. X. In the I'reu. ; Ο -^ Ml -"^ Η DAY USE '^ w ^ '■^^N DEPT. ^ VERSITY OF CUIFORNU LIBRARY Of THE UNIVERSITY OF CALIFORNIA Mm THE U ϋ ~( ΛΤ:'/ / GAUFORNm ' « T,V ^ ^M Qi^ Ο m μ^ί^ρίο THE UNIVERSITY OF CUIFHiP|l}, ., 'LIBRARr-JIf. Τ<1 ί jlN-iVERSITT OF CALIFORIN 4 OV_/f^) \ / ■:^/ QJJ ^ 7